Compactness in the metric space: Theorem: 1) X ix compact 2) Every sequence has a convergent subsequence converging to apoint in K. 3)Then K is closed and bounded. I just want you to prove the 2) => 3) in metricspace. For example I am showing that 1) =>2) By contradiction there exist sequence subset K with noconvergent subsequence. For all xn in the sequence there existB(xn , rn) so that B(xn ,rn) contains only finitely many xj\'s. By hypothesis, 2 finitely many such balls cover K.. Only finitely many xj\'s are covered. There are infinitely many xj\'s. This is a contradiction. For example 2) => 1) Let { U } be an open cover of k. Claim:- There exist >0 for all x belongs tok B(xn, 1/n) subset to U for some. proof of claim:- Suppose not. Let (xn) be a sequence such that B(xn,1/n) not subset to U for any . by 2) there exist U such that B(a,r) subset toU. since xn--->aeventually (xn, a) < r/2 futhermore eventually 1/n < r/2 Let both these happen and m be big enough. ie,(xm, a) < r/2 and 1/m < r/2. So let Z belongs to B(xm, 1/m) (Z,a) (Z,xm)+(xm, a) < 1/m+r/2 < r/2+r/2 =r So B(xm, 1/m) subset to B (a,r) subset toU. Choose a sequence a points yn such that the ballsB(yn, /2) are disjoint. Note: This cannot be an infinite because noticeB(xn,xm) /2. So { yn } does not converge, nor do anysubsequence. By hypothesis all sequence have a convergentsubsequence. Compactness in the metric space: Theorem: 1) X ix compact 2) Every sequence has a convergent subsequence converging to apoint in K. 3)Then K is closed and bounded. I just want you to prove the 2) => 3) in metricspace. For example I am showing that 1) =>2) By contradiction there exist sequence subset K with noconvergent subsequence. For all xn in the sequence there existB(xn , rn) so that B(xn ,rn) contains only finitely many xj\'s. By hypothesis, 2 finitely many such balls cover K.. Only finitely many xj\'s are covered. There are infinitely many xj\'s. This is a contradiction. For example 2) => 1) Let { U } be an open cover of k. Claim:- There exist >0 for all x belongs tok B(xn, 1/n) subset to U for some. proof of claim:- Suppose not. Let (xn) be a sequence such that B(xn,1/n) not subset to U for any . by 2) there exist U such that B(a,r) subset toU. since xn--->aeventually (xn, a) < r/2 futhermore eventually 1/n < r/2 Let both these happen and m be big enough. ie,(xm, a) < r/2 and 1/m < r/2. So let Z belongs to B(xm, 1/m) (Z,a) (Z,xm)+(xm, a) < 1/m+r/2 < r/2+r/2 =r So B(xm, 1/m) subset to B (a,r) subset toU. Choose a sequence a points yn such that the ballsB(yn, /2) are disjoint. Note: This cannot be an infinite because noticeB(xn,xm) /2. So { yn } does not converge, nor do anysubsequence. By hypothesis all sequence have a convergentsubsequence. Let { U } be an open cover of k. Claim:- There exist >0 for all x belongs tok B(xn, 1/n) subset to U for some. proof of claim:- Suppose not. Let (xn) be a sequence such that B(xn,1/n) not subset to U for any . by 2) there exist U such that B(a,r) subset to.