Question 13 (1 point) For any bounded and nonempty subset X CR, we have sup(-2X) =...
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y of R such that (Y, dy) contains nonempty closed and bounded subsets which are not compact (here dy is the metric inherited from the usual metric in R)
Using only the definition of compact sets in a metric space, give examples...
2. If S:= {1/n - 1/min, me N}, find inf S and sup S. 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : S ES). Prove that inf(as) = a infs, sup(as) = a sup S. (b) Let b <0 and let b = {bs : S € S}. Prove that inf(bs) = b supS, sup(bs) = b inf S. 6. Let X be a nonempty set and...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
REAL ANALYSIS Question 1 (1.1) Let A be a subset of R which is bounded above. Show that Sup A E A. (1.2) Let S be a subset of a metric space X. Prove that a subset T of S is closed in S if and only if T = SA K for some K which is closed in K. (1.3) Let A and B be two subsets of a metric space X. Recall that A°, the interior of A,...
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Question 13 (1 point) How many critical numbers does the function f(x) = 2x - 3x2 + 4 have? 3 2 0 Question 14 (1 point) The vertical asymptotes of X are f(x) = O x=2 and x = 1 Oy= 0 and x = 2 x = 2 and x = -2 x= 0 and x = 2
Q2 Question 2 1 Point If Q is a rectangle and f:Q → Ris nonzero only on a closed subset of measure zero, then So f = 0. true false Save Answer Q3 Question 3 1 Point Let f : [0,1] × [0, 1] → R be a bounded function such that the integrals So So f(x)dydx and So So f(x)dxdy both exist and are equal. Then S10,1)*(0,11 f exists. true false Save Answer
Question 13 1 pts TRUE OR FALSE PRINGLE???? The point (-1, -1) is a saddle point for the function f(x, y) = x2 – 3y2 + 2(x – y). O True O False
CR, we typically think of t if : >0.. 1-1 if : <o'' this is the natural way we might define the 'magnitude of a real number, but it is not the only way. a.) Prove that for ry ER, we have xy = 13. lyl. b.) Construct a new function : R-R UO) such that for r, y € R, we have: 1.) ||2||=0- I = 0 and ii.) ||3+ yll |||| + llyll but iii.) xyll ||||llyll. 36....
Evaluate: vr y-x dA , y + 2x+1 where R is the parallelogram bounded by y-x-2, y-x-3, y + 2x = 0, andy+2x=4.
Evaluate: vr y-x dA , y + 2x+1 where R is the parallelogram bounded by y-x-2, y-x-3, y + 2x = 0, andy+2x=4.