The following is a useful characterisation of convergence for sequences in $\mathbb R$:

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> A sequence $(a\_n)$ converges to $a$ if and only if every subsequence $(a\_{n\_k})$ of $(a\_n)$ has a subsequence $(a\_{n\_{k\_\ell}})$ converging to $a$. 
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The usual proof of this statement is very short. If $(a\_n)$ converges to $a$, it is immediate that every subsequence does so, and each of their subsequences also converges to $a$. Conversely, suppose $(a\_n)$ does not converge to $a$. This means there exists $\varepsilon > 0$ such that for every $N > 0$, there exists $n > N$ with $|a\_n - a| \geq \varepsilon$. These $a\_n$ for a subsequence with no further subsequences converging to $a$, so by the contrapositive, we are done.

However, the second part of the proof says that the "negation is false" rather than saying that "the result is true". Proving that the negation is false is easy because we can rather concretely construct a subsequence that fails, but my attempts to give a direct proof have never been fruitful. Does anyone know any direct proofs of this fact?