Let an and br be sequences. Prove that sequences an and by both converge iff both...
nees in (Mod) that converge to (2) Let (an) and (y) be sequences in & and y respectively. Prove that d(x,y) d(x,y). If SCM is bounded, prove that the diameter o the diameter of cics) are equal. I the diameter ofs and
Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge. 2Ti3 1
Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge. 2Ti3 1
for every n. Prove: If (a) converges, then 11. Let (a.) and (b) be sequences such that a, b, < so does (bn). There are several ways to prove this; at least one doesn't involve Cauchy sequences or e. Be careful though you don't know that () converges so make sure that your method of proof doesn't in fact require (b) to converge.
(3) IF THE SEQUENCES {x, + y} AND {X- BOTH CONVERGE THEN THE SEQUENCES (X) and ty} ARE BOTH CONVERGENT. AND CONVERSELY
Provide an ? N proof to prove that the following sequences
converge.
Question (e), please.
5. Provide an e – N proof to prove that the following sequences converge. (a) {ne cos(n)} (b) {zo Bom} (c) {(-1)In (n)} (d) an = 2 + 1 (@) an = V1 -
1. Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge n-+2 (b) an= n-cos(n) 4n2-7 Tn (d) { } 2. Prove that the following sequences diverge. (Def 7.10 pg 596) READ Sequences that Diverge to oo or-oo (b) ann infinity. Hint: Provide an M -N proof that an approaches
1. Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge n-+2 (b) an=...
Prove the following
Definition 6.6.1 (Subsequences). Let (an) =) and (bn), m=0 be sequences of real numbers. We say that (bn)is a subsequence of (an) a=iff there exists a function f :N + N which is strictly increasing (i.e., f(n + 1) > f(n) for all n EN) such that bn = f(n) for all n E N.
1. Provide a complete and accurate e- N proof that the following sequences converge. That is, prove these sequences converge. n2+2 (b) an- 30s n-cos(n) 3n+2 e) an-4m2-7 (d) {
Can you solve No.6
6. Let (a.)and b) be bounded sequences in R .a. Prove that lima. +İimb, siim (a, + b.) s ima, + İim br b. Prove that lim (-a)lima . Given an example to show that equality need not hold in (a) If o, and b, are positive for all n, prove that lim (a)s(im a)mb). provided the product on the right is 7. not of the form 0 oo. b. Need equality hold in (a)?
6....
, { x,1. {%) and {4) hat converge almost surely to a,b,c onsider three sequences of random variables respectively. Prove that a.s
, { x,1. {%) and {4) hat converge almost surely to a,b,c onsider three sequences of random variables respectively. Prove that a.s