Does my alternative proof of Bolzano-Weierstraß make sense, or am I missing something?

Question

I want to show that every bounded sequence in $\mathbb{R}$ has a convergent subsequence. Let $a_n$ be any bounded sequence in $\mathbb{R}$. Since $a_n$ is bounded, $\text{limsup}(a_n)$ exists, call it $k$. By lemma, for any $\epsilon > 0$, the set $\{n \;|\; \text{limsup}(a_n) - \epsilon < a_n < \text{limsup}(a_n) + \epsilon \}$ is infinite. If we let $\epsilon$ become arbitrarily small, and include only the $n$'s from our defined set into our subsequence, we have obtained a subsequence, that, by definition, converges to $k$. Hence, every bounded sequence in $\mathbb{R}$ has a convergent subsequence.