Question about a proof of Bolzano-Weierstrass theorem

Your explanation doesn't seem quite right to me because when you write $a_m=a_n+L\leq -\frac{1}{k}$ the variable $n$ is undefined. Do you mean that there exists $n$ such that this inequality is true? Or that it is true for all $n$? Or something different?

The way to justify that $H_k^+$ is infinite is to note that $L+\frac{1}{k}>L$ for every $k$, and since $L$ is the infimum of $X$ there is some $x_0\in X$ such that $x_0<L+\frac{1}{k}$ (otherwise, $L+\frac{1}{k}$ would be a lower bound of $X$ which is bigger than $L$). By definition of $X$ this means that $\{n\in\mathbb{N}:a_n\leq x_0\}$ is infinite, and since $x_0<L+\frac{1}{k}$ this means that $\{n\in\mathbb{N}:a_n\leq L+\frac{1}{k}\}$ is infinite, and this is just $H_k^+$. It should go similarly for $H_k^-$.

(I have to add that it seems to me that the notation used in this proof is likely to be quite confusing for someone who is supposed to be seeing a proof of Bolzano-Weierstrass for the first time.)