= Monotone subsequences characterize convergence
{title2=$a_n\to L\iff\text{every monotone subsequence tends to }L$}
For a <sequence> of <real numbers>, \b[a common finite <limit of a sequence> for all <monotone subsequences> forces the whole <sequence> to have that <limit of a sequence>.] There is no need to assume a <bounded sequence> if the hypothesis already includes convergence of every <monotone subsequence>.
If $a_n$ fails to tend to $L$, select a <subsequence> with $|a_{n_j}-L|\geq\varepsilon$ for some fixed $\varepsilon>0$. The <monotone subsequence theorem> gives a <monotone subsequence> of this <subsequence>, which contradicts the common <limit of a sequence> $L$. Conversely, every <subsequence> of a <convergent sequence> has the original <limit of a sequence>.
For a <bounded sequence>, the <bounded monotone sequence theorem> already ensures that all <monotone subsequences> converge. Only agreement of their <limits of sequences> remains to be checked.
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