Question: Let S = {x: x is rational and x2 < 2}. Convince yourself that S does not have a least upper bound in the rational

Let S = {x: x is rational and x2 < 2}. Convince yourself that S does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence of rational numbers 1, 1.4, 1.41, 1.414,...., has no limit within the rational numbers?

Step by Step Solution

★★★★★

3.32 Rating (158 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Recall that every rational number can be written as either a terminating or ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

955-M-C-D-E (2453).docx

120 KBs Word File

Students Have Also Explored These Related Calculus Questions!

Let f : R R be defined by setting f(x) := x if x is rational, and f(x) = 0 if x is irrational. (a) Show that f has a limit at x = 0. (b) Use a sequential argument to show that if c 0, then f does...

Let f(x) = 0 if x is irrational and let f (x) = 1/q if x is the rational number p/ q in reduced form (q > 0). (a) Sketch (as best you can) the graph off on (0, 1). (b) Show that f is continuous at...

In each of the following problems, plot the graph of f(x, y) and the corresponding gradient field F = (f on S = {(x, y)}; - 3 ( x ( 3, - 3 ( y ( 3}. In each case, curl F = 0 (Theorem 14.3D) and so...

QUIZ... Let D be a poset and let f : D D be a monotone function. (i) Give the definition of the least pre-fixed point, fix (f), of f. Show that fix (f) is a fixed point of f. [5 marks] (ii) Show that...

Q10 (4 points) Let S be a bounded set of real numbers and suppose that Sup S =Inf S. What can you conclude about S ? Drag and drop an image or PDF file or click to browse Q11 (6 points) Show by...

[3.13] Answer the following questions along with a concise explanation with respect to the linear program to maximize cx subject to xX={x:Ax=b,x 0}, where A is mn of rank m 0, then is the optimal...

1. The following proof has an error. Find the error that makes the proof incorrect. Claim 1 . The set [0, 1] has no least upper bound. Proof . For sake of contradiction, let a = sup[0, 1]. Case 1 ,...

Recall that in class we did most of the proof that Q does not have the least upper bound property. We considered the set S = {r Q| r 2

The purpose of this exercise is to prove the following Vapnik-Chernovenkis bound: for any finite class $\mathscr{G}$ (containing only a finite number $|\mathscr{G}|$ of possible functions) and a...

Please show all the steps, and state theorems use. Let S be a nonempty bounded set containing a sequence of rational numbers. Provide an example such that the least upper bound of S is not a rational...

Nueva Company reported the following pretax data for its first year of operations. Net sales Cost of goods available for sale 7,540 5,680 1,578 Operating expenses Effective tax rate Ending...

There are 31 pairs of spinal nerves. How many pairs of spinal nerves are there in each section? Just give the numeral, don't write out the word for the number. Cervical: Thoracic: Lumbar: Sacral:...

19. Rapel Valley in Chile is renowned for its ability to produce high-quality wine at a fraction of the cost of many other vineyards around the world. Rapel Valley produces over 20 million bottles of...

In panel (b) of Figure 12.9 on page 484, Sacha Gillette reduces her output from 40,000 to 25,000 dozen eggs when the price falls to $1.75. At this price and this output level, she is operating at a...

Calculate the sum of the series whose partial sums (n=1 an whose partial sums are given. Sn = 2 - 3(0.8) n

Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Example 7). If it is convergent, find its sum. a. b. c. -i nn + 3)

Let x = 0.99999 .... (a) Do you think that x

STATS 134-B READ COMMENTS FOR INSTRUCTIONS! PLEASE SKIP IF YOU CANT FINISH! I WILL REPORT THOSE WHO WILL FLAG THIS. THANKS NEED URGENT! 6. When attributes charts are used when _______ a) Destructive...

3.17 Insurance company losses of a particular type are to reasonable approximation normally distributed with a mean of $150 million and standard deviation of $50 million. (Assume no different between...

For each of the following annuities, calculate the annual cash flow For each of the following annuities, calculate the annual cash flow. (Enter rounded answers as directed, but do not use rounded...