## Question

###### 1- Complete the following proof. Theorem Let S ER: Let p â‚¬ R The following two statements are equitlent: (i) Ve > Ve(p)o (S| {p}) "onemptu: (ii) P= lim dn for some sequence (an) contained in $ | {p} (Recall that if any of the above equivalent conditions holds, then we say that is a limit poinf of $.) Proof: (ii) Let â‚¬ be given_ Qur goal , is to show that Ve(p)n(S| {p}) is nonempty: We have Uj `p JN â‚¬N Vn >N dn â‚¬ Velp) In particular; aN+I Ve(p) . Also notice that; by assumpti

1- Complete the following proof. Theorem Let S ER: Let p â‚¬ R The following two statements are equitlent: (i) Ve > Ve(p)o (S| {p}) "onemptu: (ii) P= lim dn for some sequence (an) contained in $ | {p} (Recall that if any of the above equivalent conditions holds, then we say that is a limit poinf of $.) Proof: (ii) Let â‚¬ be given_ Qur goal , is to show that Ve(p)n(S| {p}) is nonempty: We have Uj `p JN â‚¬N Vn >N dn â‚¬ Velp) In particular; aN+I Ve(p) . Also notice that; by assumption aN+1 â‚¬s {p}: This shows that Ve(p)n(s| {p}) is nonempty: (complete the proof. Hint: Notice that for all â‚¬N; by assumption; (p- #p+4) has nonempty intersection with {p} Construct sequence (dn) by letting On be point in the nonempty intersection of (p 2P+ and {p}- Use the Squeeze Theorem to show that indeed &r - 2- Let sequence of real numbers. Let s = {an : " â‚¬N} Prove that if L is limit point of (that is, if L â‚¬ $'), then there exists subsequence of (an) whose limit is L Proof: In the reading assignment posted to the course homework page we proved the following lemma: Lemma. Let E CR Let p â‚¬ E' and suppose that 0 is aJ open set containing Tlen On_ contains infinitely aiI pomts Since L â‚¬ $', it follows from the above lemma that (L-LLtI)nshas infinitely many points 3n such that dnp (L-LL+I) L+ ins has infinitely many points 302` "g such that arz â‚¬ (L = ~L+ Lt nshas infinitely many points 3n3 nz such that Ors 3-L+ In ths way we obtain subsequence Un such that VkeN Sam < Lt} (complete the proof. Hint: Use the Squeeze Theorem to show that lim Ont = L)