(2) Define the set AC by A -{int el: n-0 (d) Prove that A is compact. (2) Define the set AC by A -{int el: n-0 (d...
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact (2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
(2) Define the set A by (a) Prove that for any N 20 the set is compact. (b) Prove that for any e>0 there exists some N 2 0 so that for any x A we have (c) Prove that A is totally bounded. d) Prove that A is compact.
the set A ⊆ L^2 by A = { {xn} ∈ L^ 2 : X∞ n=0 (1 + n)|xn| 2 ≤ 1 } Prove A is totally bounded, and compact.
Prove that in R^n with the usual topology, if a set is closed and bounded then it is compact.
5. Let A and B be compact subsets of R. (a) Prove that AnB is compact (b) Prove that AUB is compact. (c) Find an infinite family An of compact sets for which UAn is not compact. o-f (d) Suppose that An is a compact set for n 21. Prove that An is compact.
1. (a) Prove that a closed subset of a compact set is compact. (b) Let a, b € R and f: R → R, x H ax + b. Prove that f is continuous. Is f uniformly continuous?
2. Consider the set S-[1, oo). Consider the open cover x(n-1,n+)InEN) - (0,2),(1,3),(2.4),(8,5.,..) of S. Prove that X contains no finite subcover of S. Hence S is not compact. 2. Consider the set S-[1, oo). Consider the open cover x(n-1,n+)InEN) - (0,2),(1,3),(2.4),(8,5.,..) of S. Prove that X contains no finite subcover of S. Hence S is not compact.
Prove that the convex hull of a set using the fact that it is compact. x1,., nin R" is bounded,, without Prove that the convex hull of a set using the fact that it is compact. x1,., nin R" is bounded,, without
Im wondering how to do b). (6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...