Real Sequences Lecture 7

This is the final lecture in this playlist: We explain the uses of subsequences. We then state and prove the sequential version of Bolzano-Weiesrtrass theorem. This leads to a motivated definition of a cluster point of a set and cluster point version of B-W theorem. We then give a standard proof of Cauchy completeness of R. We end up with the notion of sequences diverging to infinity and minus infinity. We also point out the difference between unbounded sequeneces and those that diverge to plus or minus infinity. 00:00 Start 06:40 Uses of Subsequences 20:43 Bolzano Weierstrass Theorem (Sequential version) 28:20 Bolzano Weierstrass Theorem (Set Theoretic version) 39:40 Cluster / Accumulation Point 48:00 Every Cauchy Sequence in R is convergent 52:05 Sequences diverging to infinity and minus infinity 1:05:45 n-th root of (n!) diverges to infinity 1:16:20 n times (-1)power n is unbounded but neither diverges to plus infinity nor minus infinity 1:19:03 Some advice to learn analysis