5.6.8 let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e). show that this need not be true if f is continuous but not uniformly continuous.
Proof. (1): To prove {f(xn)} is a Cauchy sequence just need to prove ∀ > 0, ∃N, s.t.,∀n, m > N,
have |f(xn) − f(xm)| < . Since f is uniformly continuous on set E, thus ∀ > 0, ∃δ > 0, s.t.,
∀x, y ∈ E, if |x − y| < δ,then |f(x) − f(y)| < .as {xn} is a Cauchy sequence, then ∃N, s.t.,
∀n, m > N, |xn − xm| < δ, thus |f(xn) − f(xm)| < which proves {f(xn)} is a Cauchy sequence.
(2): for example f(x) = 1
x
, x ∈ (0, 2) which is continuous but not uniformly continuous. {
1
n
} is
a Cauchy sequence, however, {f(xn)} does not converge which proves that it is not a Cauchy
sequence.
Answers
Showing that this need not be true if f is continuous but not uniformly continuous.
Given :
let f be a uniformly continuous function on a set e. show that if {xn} is a cauchy sequence in e then {f(xn)} is a cauchy sequence in f(e).
( 1 )
If f is uniformly continuous on E, then given ε > 0 there is δ > 0 such that if x, y are in E and |x−y| < δ,
then |f(x) − f(y)| < ε. Let (xn) be a Cauchy sequence in E. Then given δ > 0 there is N such that if p, q > N,
then |xp − xq| < δ, and thus |f(xp) − f(xq)| < ε, implying that (f(xn)) is a Cauchy sequence.
( 2 )
Let E = {1, 1/2, 1/3, · · } and f(1/n) = 1 is n is odd, f(1/n) = −1 if
n is even. Then f is continuous but not uniformly continuous. The sequence (xn) = (1/n) in E is Cauchy but the
sequence (f(xn)) = (1, −1, 1, −1, · · ·) is not Cauchy.
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Related Questions
The cost to mail a package is $7 for the first 2 pounds and 30 cents for each additional ounce.
Which of the following functions represents the cost to mail a package if x is the number of ounces over 2 pounds?
ƒ(x ) = 7 + 0.3x
ƒ(x ) = 14 + 0.3x
ƒ(x ) = 0.3(x + 2)
Answers
Answer:
the first one
Step-by-step explanation:
The cost to mail a 2-lb package is $7.
The cost to mail a 2-lb, 1 oz package is $7+$0.30(1).
That to mail a 2-lb, 2 oz package is $7+$0.30(2) = $7.60.
Following this pattern, the general formula is f(x) = $7 + $0.30x, where x represents the number of ounces OVER 2 lb.
The 8th grade math team is having a bake sale as a fundraiser. Alan can make 5 cookies in 2 minutes. Abigail can make 8 cookies in 3 minutes. However, Katie eats 1 cookie every 6 minutes. How many minutes will it take to make 5 dozen cookies.
Answers
Answer: B. 12
Step-by-step explanation:
1) Alan can make 2.5 cookies per min
2) Abi can make 8/3 which is 2.7.
3) overall, 5.2 cookies per minute can be made.
multiple 5.2 x 6 and you get around 31.5?
then subtract =1. you have 30.5, now add onto that number 31.5, which is another 6 minutes. since you're still only subtracting 1, the answer would be above 60, which is 5 dozen, and it only took 12 mins
Verizon Wireless would like to estimate the proportion of households that use cell phones for their phone service without a land line. A random sample of 150 households was selected and 48 relied strictly on cell phones for their service. The margin of error for a 90% confidence interval for the proportion based on this sample is ________.
Answers
Answer:
\( ME = 1.64 \sqrt{\frac{0.32 (1-0.32)}{150}}= 0.0625\)
Step-by-step explanation:
For this case we have the following info given:
\( n = 150\) represent the sampel size selected
\( X = 48\) represent the number of households who relied strictly on cell phones for their service
The estimated proportion of households who relied strictly on cell phones for their service is given by:
\( \hat p =\frac{X}{n}= \frac{48}{150}= 0.32\)
And the margin of error would be given by:
\( ME = z_{\alpha/2} \sqrt{\hat p(1-\hat p)}{n}\)
The confidence is 90% so then the significance is \(\alpha=1-0.9=0.1\) and \( \alpha/2 =0.05\) the critical value for this case from the normal standard distribution is:
\( z_{\alpha/2}= 1.64\)
And the margin of error would be:
\( ME = 1.64 \sqrt{\frac{0.32 (1-0.32)}{150}}= 0.0625\)
Find the height of the tower using the information given in the illustration.
Answers
using SOH CAH TOA
Tan 85.144 =h/130
h=tan 85.144*130
h=1530.19 fr
HELP WILLING TO CASHAPP if answered quick and correctly + 100 points The points plotted below on the graph of a polynomial which of the following X values is the best approximation of a root of the polynomial? Check all that apply
X=-0.21
X=-4.6
X=-2.3
X=3
X=1
X=-3.5
Answers
first 1
4th one
i think sorry if its wrong
Answer:
Step-by-step explanation:
Did you ever find out the answer????
HELP PLZZZ. In an indirect proof, you prove an if- then statement is true by assuming the statement is false then disproving the false statement. You want to prove the statement below in an indirect proof. What statement should you prove false.
Statement to prove true: if a figure has exactly three sides,then it is a triangle.
Statment to prove false:???
Answers
Answer:
If a figure is a triangle, then it has three sides.
Step-by-step explanation:
You just flip the if then statement around.
PLEASE HELP ME :(. Look at this system of equations.
X - 8y = -3
5x - 2y = 4
What value of a will make this system have the same solution?
ax = -19
5x - 2y = 4
Answers
9514 1404 393
Answer:
a = -19
Step-by-step explanation:
In the first set of equations, subtract 4 times the second equation from the first.
(x -8y) -4(5x -2y) = (-3) -4(4)
-19x = -19 . . . . simplify
Comparing this to the equation ...
ax = -19
we see that they will have the same solution when a = -19.
In Steven's grade, 12 students are enrolled in health and 48 students are not. What percentage of the students in the school are enrolled in health?
Answers
Answer:
20%
step-by-step way
the radius of a cylindrical construction pipe is 3.5. the pipe is 34 ft long what is it's volume
Answers
Answer:
1308 ft³.
Explanation:
For a cylinder with radius r and height, h:
\(\text{Volume}=\pi r^2h\)Given:
• Radius. r= 3.5 ft
,• Length, h = 34 ft
,• π = 3.14
\(\begin{gathered} \text{Volume}=3.14\times3.5^2\times34 \\ =1307.81ft^3 \\ \approx1308ft^3(to\text{ the nearest whole number)} \end{gathered}\)The volume of the pipe is 1308 cubic feet.
please help me please help me please
Answers
Answer:
Pull.
Step-by-step explanation:
Because the person is bringing the cart closer to himself, it is a pill force.
Hope this helps
Please answer the question in the photo
"What is the value of x in the figure?"
(x+10)
------------------
30
Answers
x+40=90
x=50
Washington, DC is 389 miles from Statesville, NC. If you wanted to drive there,
how long would it take you driving on interstates with an average of 65 mph?
O 5.98 hours
07.07 hours
O 5.56 hours
07.78 hours
O 6.48 hours
Suppose your car gets 29 miles per gallon on the interstate and gas costs
$3.89/gallon. How much will it cost you to drive to Washington, DC?
O $0.29
O $43,883.09
$52.18
O $3.45
O $2,900.00
Answers
It would cost approximately $52.18 to drive from Statesville, NC to Washington, DC with a car that gets 29 miles per gallon on the interstate, considering a gas cost of $3.89 per gallon.
To calculate the time it would take to drive from Statesville, NC to Washington, DC, we can divide the distance of 389 miles by the average speed of 65 mph.
Time = Distance / Speed
Time = 389 miles / 65 mph
Time ≈ 5.98 hours
Therefore, it would take approximately 5.98 hours to drive from Statesville, NC to Washington, DC on interstates with an average speed of 65 mph.
To calculate the cost of driving to Washington, DC, we need to know the number of gallons of gas required for the trip. We can find this by dividing the distance by the car's mileage, which is 29 miles per gallon.
Number of gallons = Distance / Mileage
Number of gallons = 389 miles / 29 miles per gallon
Number of gallons ≈ 13.41 gallons
The cost of gas can be calculated by multiplying the number of gallons by the cost per gallon, which is $3.89.
Cost = Number of gallons * Cost per gallon
Cost ≈ 13.41 gallons * $3.89/gallon
Cost ≈ $52.18
Therefore, it would cost approximately $52.18 to drive from Statesville, NC to Washington, DC with a car that gets 29 miles per gallon on the interstate, considering a gas cost of $3.89 per gallon.
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from a group of 9 boys and 13 girls, a committee of 5 students is chosen at random. a. what is the probability that all five members on the committee will be girls? b. what is the probability that all 5 members of the committee will be boys?c. what is the probability there will be at least 1 girl on the committee?
Answers
Answer
a) Probability that all five members on the committee will be girls = (143/2926) = 0.0489
b) Probability that all five members on the committee will be boys = (1/209) = 0.00478
c) Probability there will be at least 1 girl on the committee = (208/209) = 0.995
Explanation
The probability of an event is calculated as the number of elements in that event divided by the total number of elements in the sample space.
For this question,
Total number of ways we can select a committee of 5 students from 9 boys and 13 girls = 22C5 = ²²C₅ = 26,334
a) Number of ways of selecting only girls in the committee of 5 = 13C5 = ¹³C₅ = 1,287
Probability that all five members on the committee will be girls = (1287/26334) = (143/2926) = 0.0489
b) Number of ways of selecting only boys in the committee of 5 = ⁹C₅ = 126
Probability that all five members on the committee will be boys = (126/26334) = (1/209) = 0.00478
c) Probability there will be at least 1 girl on the committee
= 1 - (Probability that there will be no girl on the committee)
= 1 - (Probability that there will only boys on the committee)
= 1 - (1/209)
= (208/209)
= 0.995
A group of friends really love to go bowling together. Here are a set of final scores from a recent bowling trip:
147, 160, 80, 110, 121, 164, 62, 90, 76, 83, 110
Required:
What is the mean?
Answers
Answer:
\(Mean = 109.364\)
Step-by-step explanation:
Given
\(Data: 147, 160, 80, 110, 121, 164, 62, 90, 76, 83, 110\)
Required
Calculate the mean
Mean is calculated as:
\(Mean = \frac{\sum x}{n}\)
In this case:
\(n = 11\)
So, we have:
\(Mean = \frac{147+ 160+ 80+ 110+ 121+ 164+ 62+ 90+ 76+ 83+ 110}{11}\)
\(Mean = \frac{1203}{11}\)
\(Mean = 109.364\) -- approximated
You invest some money made a profit of $75. Your profit was 10% of your investment. How much did you invest originally?
Answers
Answer:
85
Step-by-step explanation:
may be thats the aNswer
Convert the following polar-form vectors of the form ⟨r,θ⟩ into component-form vectors of the form ⟨x,y⟩. Your answers should be in the form "< #, # >". ⟨13,50∘⟩= ⟨1.77,250∘⟩= ⟨5.8,150∘⟩=
Answers
9514 1404 393
Answer:
a) <8.356, 9.959>
b) <-0.605, -1.663>
c) <-5.023, 2.9>
Step-by-step explanation:
Many calculators can perform polar ⇔ rectangular conversion. Attached is the result from one of them. Of course, you can also program a spreadsheet to do it. (The ATAN2( ) function is useful for finding the correct angle.)
If you want to do these calculations by hand, the conversion is ...
<r, θ> ⇒ <r·cos(θ), r·sin(θ)>
In the attached, the rectangular coordinates are shown as complex numbers. The imaginary component is the y-component of the vector.
Are the triangles similar? If so, what postulate or theorem proves
their similarity?
Answers
Answer:
Yes, by SSS
Step-by-step explanation:
Reflections of Shapes
Graph the image of the figure using the transforma
1) reflection across the x-axis
Answers
A graph of the image of the figure after a reflection across the x-axis is shown below.
What is a reflection over the x-axis?In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).
This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.
By applying a reflection over or across the x-axis to triangle GLQ, we have;
(x, y) → (x, -y)
G (3, 4) → (3, -(4)) = G' (3, -4)
L (1, 2) → (1, -(2)) = L' (1, -2).
Q (4, -1) → (4, -(-1)) = Q' (4, 1)
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Addison has x pennies and y nickels. She has a minimum of 16 coins worth no more than $0.40 combined. Solve this system of inequalities graphically and determine one possible solution.
Inequality 1
Inequality 2
Answers
From the graph, a possible solution to the inequalities x + y ≥ 16 and 0.01x + 0.05y ≤ 0.4 is (15, 2)
What is an equation?An equation shows how two or more numbers and variables are related to each other.
Let x represent the number of pennies and y represent the number of nickels. One penny = $0.01 and one nickel = $0.05
She has a minimum of 16 coins, hence:
x + y ≥ 16 (1)
Also:
It is no more than $0.40 combined, hence:
0.01x + 0.05y ≤ 0.4 (2)
From the graph, a possible solution is (15, 2)
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Point A is translated 4 units left and 3 units up. What are the coordinates of point A in its new position ?
Answers
Answer:
A (-4,3)
Step-by-step explanation:
Answer:
(-4,3)
Step-by-step explanation:
know that left and right is the x axis we start from there. left is negative right is positive, so I know 4 unit left is -4.
knowing that up and down is the y axis and y axis always comes second. up is positive down is negative. so I know that 3 units up is positive.
thats how we got( -4,3)
Rashau made a rectangular frame for her latest oil painting. The length is 18 centimeters more than double the width.
The perimeter of the frame is 96 centimeters. Find the length and the width of the frame.
Width = cm
CONT
200
Answers
Answer:
width: 10 centimeters
length: 38 centimeters
Step-by-step explanation:
Let's start by assigning variables to the width and length of the frame.
Let x be the width of the frame in centimeters.
Then, according to the problem statement, the length of the frame is 18 centimeters more than double the width, which can be expressed as 2x + 18.
The perimeter of a rectangle is the sum of the lengths of all four sides, so the perimeter of this frame is:
2(length + width) = 2(2x + 18 + x) = 6x + 36
We know that the perimeter of the frame is 96 centimeters, so we can set up an equation:
6x + 36 = 96
Simplifying this equation, we get:
6x = 60
x = 10
So the width of the frame is 10 centimeters.
To find the length, we can plug x = 10 into the expression we found earlier for the length:
length = 2x + 18 = 2(10) + 18 = 38
So the length of the frame is 38 centimeters.
Therefore, the width of the frame is 10 centimeters and the length is 38 centimeters.
6 inches every 4.5 days to yards per year
Answers
Explanation: you would do 365/4.5
Then that times 6 then divide by 36
A group of friends wants to go to the amusement park. They have $114.75 to spend on
parking and admission. Parking is $8.75, and tickets cost $26.50 per person,
including tax. Which equation or tape diagram could be used to represent the context
if a represents the number of people who chan go to the amusement park
Answers
The equation if a=4 represents the number of people who can go to the park is 8.75 + 26.50a = C.
What are equations?A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.
a formula that expresses the connection between two expressions on each side of a sign.
Typically, it has a single variable and an equal sign.
Like this: 2x - 4 = 2.
In the above example, the variable x exists.
So, the equation represents the number of people who can go to the park:
The total budget is $114.75.
Parking costs $8.75.
The cost of a ticket per person is $26.50.
So form the equation:
Let, a be the number of people and C be the total cost.
8.75 + 26.50a = C
Now, let a be 4.
8.75 + 26.50a = C
8.75 + 26.50(4) = C
8.75 + 106 = C
114.75 = C
Therefore, the equation if a=4 represents the number of people who can go to the park is 8.75 + 26.50a = C.
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K
At a computer store, a customer is considering 8 different computers, 7 different monitors,
9 different printers and 2 different scanners. Assuming that each of the components is
compatible with one another and that one of each is to be selected, determine the number
of different computer systems possible.
Answers
there are 1,008 different computer systems possible, given the constraints provided.
what is constraints?
Constraints are limitations or conditions that must be satisfied in order to solve a problem or make a decision. In other words, constraints are rules or restrictions that define the boundaries within which a solution or decision must be made.
In the given question,
To determine the number of different computer systems possible, we need to multiply the number of choices for each component. This is based on the multiplication principle of counting, which states that if there are m ways to do one thing and n ways to do another thing, then there are m x n ways to do both things together.
So, for this problem, we have:
8 different choices for computers
7 different choices for monitors
9 different choices for printers
2 different choices for scanners
Using the multiplication principle, the total number of different computer systems possible is:
8 x 7 x 9 x 2 = 1,008
Therefore, there are 1,008 different computer systems possible, given the constraints provided.
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may you solve this for me?
Answers
Answer: 135 m
Step-by-step explanation: In order to find the area, you should split the shapes in 2. The 2 shapes are a trapezoid and a rectangle. To find the area of the rectangle, you need to multiply the length and width, which is 18 * 5, which is 90. For the trapezoid, you need to add the top and bottom part of the trapezoid, which is 12 + 18, then multiply it by the height divided by 2. The height is 3 because the total height is 8 and some of the height is 5, so the remaining height is 3, which is the trapezoid's height. Then, divide 3 by 2, which is 1.5. Then multiply 1.5(12+18), which is 45. Add the area of both shapes together, 45 + 90, and it is 135.
:)
Which set of data is displayed in the box plot below?
Answers
Answer:
It would tske to long for me to break it down but my best bet is B
Step-by-step explanation:
the value of square root (8/64)³
Answers
The expression is
\(\begin{gathered} (\sqrt[]{\frac{8}{64}})^3 \\ By\text{ simplifying, we have} \\ (\sqrt[]{\frac{1}{8}})^3 \\ =\text{ (}\frac{1}{8})^{\frac{3}{2}} \\ 0.0442 \end{gathered}\)
PLEASE HELP!!!!
Which angle measure must equal 52°?
A. а
в. Ь
с. с
D. d
Е. е
Answers
Answer:
Option C will be your answer
Step-by-step explanation:
Angle C will be equal to 52 by vertical angle theorem
hope it helps you
have a great day !!
Angle c is the vertical opposite angle with 52° so it will have a measure of 52° thus option (C) is correct.
What is an angle?An angle is a geometry in plane geometry that is created by 2 rays or lines that have an identical terminus.
The identical endpoint of the two rays—known as the vertex—is referenced as an angle's sides.
Two opposite angle made by two intersecting lines is said to be vertically opposite angle so the angle opposite to 52° is c so, ∠c = 52°.
Hence " Angle c has a measure of 52° since it is the vertical opposite angle with 52°".
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Which expression simplifies to 41? Choose all that apply!
10+ 2.4-1
10+ 2.(4 - 1)
(10 + 2). 4 - 1
10+ (22.4) - 1
Answers
Answer:
A
Step-by-step explanation:
I took the test so yeah hope this helps
Eveluate the expression for a = 5
2a + 6
Answers
Answer:-11
Step-by-step explanation:
Answer:
-11
Step-by-step explanation:
a = 5
2a + 6 = 5
Substitute 5 in the place of a.
2(5) + 6 = 5
10 + 6 = 5
16 = 5
0 = 5 - 16
= -11