We prove three major results concerning subsequences: (1) If xn converges to x, any of its subsequence also converge to x, (2) If a Cauchy sequence (xn) has a convergent subsequence converging to x, then xn converges to x and (3) If (xn_)is real sequence, there exists a monotone subsequence.
00:00 Start
00:50 Any subsequence of a convergent sequence converges to the same limit
04:20 Curry leaf trick
08:50 If a Cauchy sequence has a convergent subsequence, then the Cauchy sequence is also convergent
32:35 Every Cauchy sequence are bounded
38:25 Every sequence has a monotone subsequence
