Asequence{un}issaidtodivergetoinfinityifgivenanyK>0there exists N such that un > K whenever n > N. In this case, we write un →∞ Monotonic bounded theorem Every bounded monotonic sequence converges. Bolzano–Weierstrass Theorem Every bounded real sequence contains a convergent subsequence. ...
How to prove that every bounded sequence in \mathbb{R} has a convergent subsequence. Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit...