Question

If a sequence is bounded, then it has at least two convergent subsequences. Check whether the given statement is true of false. Give a counterexample in support of your answer.

Answer 1

Let $a_1,a_2,…$ be a bounded sequence in $\R$ that does not converge.

Let ​$s_n$ be a bounded sequence in $\R$ that does not converge. Then by the Bolzano Weierstrass theorem there exists a subsequence, $s_{n_k}$, that converges. Let a be the limit of $s_{n_k}$​​. By definition $s_n$ does not converge so it can not converge to a. That is, there exists $ϵ>0$ such that for all $N∈\R$, there exists $n>N$ such that $∣s_n​−a∣≥ϵ$. So there exists a sequence $s_m​$ that is a subsequence of $s_n​$ with $m>N$. By its definition $s_m$ does not have a subsequence that converges to a. However, $s_m$ is bounded because $s_n$ is bounded. So by the Bolzano Weierstrass theorem there exists a subsequence, $s_{m_j}$, that converges.

Clearly $s_{m_j}$ does not converge to a.

$\therefore$ The statement is Fa.lse.