Question

Let the bounded set s contains a sequence sn of real numbers

Answer 1

Let s=supSs=supS. We know that:

for any ε>0, there is x∈S such that s−ε<x.

for any ε>0, there is x∈S such that s−ε<x.

Thus for each n>0n>0 you can choose xn∈Sxn∈S such that

s−1n<xn.

s−1n<xn.

Clearly, then, s−1n<xn≤ss−1n<xn≤s for all n>0n>0, so limnxnlimnxn exists and equals ss.