## September 26, 2009

### New Proof (wonkish)

There is a lemma proposition for the Bolzano-Weierstrass Theorem, namely, a sequence converges to a number if and only if any and every subsequence of that sequence also converges to the same number (technically a trimmed-down version of this proposition, i.e. only the necessary condition part, also suffices to prove the B-W Theorem). Traditionally, this proposition, or theorem by nature, can be possibly proved with a deduction occupying one page or so. However, I discovered today that my schoolmate Ling Hao from the Math Department has a better solution to verify the sufficient condition part (from convergence of subsequences to convergence of the original sequence). He simply picks the original sequence as the selected "any" subsequence and it is obvious that this subsequence has a limit, and consequently the original sequence, which is exactly the same as the subsequence, also converges to the same limit. Proof complete. I advised he write an academic paper for this.