1 prove that every cauchy sequence is bounded 2 give a count

1) prove that every cauchy sequence is bounded

2) give a counterexemple of : every sequence has a convergent subsequence

3) suppose that x>1. prove that lim x exp(1/n) = 1

Solution

it is bounded, because (since the tag is Real-analysis): 1)The Reals are complete, so that the sequence converges to, say aa, so that, for any >0, all-but-finitely many terms are in (a,a+).

2) The terms that are (possibly) not in (a,a+)are finitely-many. A finite collection of Real numbers has an actual maximum, say M, and an actual minimum, say m.

3) All the terms of the sequence are contained in the interval (c,d), where :

c=Min{m, a} ; d=Max{M, a+}