Let $z\in \mathbb{C}$, $(z_n)_{n\geq 1} \subset \mathbb{C}$ be a sequence and $(w_n)_{n\geq 1} \subset \mathbb{C}$ a null sequence (sequence tending to zero).
Proof that: $z$ is a limit point of $(z_n)_{n\geq 1} \iff z$ is a limit point of $(z_n+w_n)_{n\geq 1}$
My attempt:
For "$\Rightarrow$":
Let $z$ be the limit point of $(z_n)_{n\geq 1}$. We know that $(w_n)_{n\geq 1}$ is a null sequence, thus $w_n\overset{n\to \infty}\longrightarrow 0$. $0$ is a neutral element regarding addition, hence $z$ is a limit point of $(z_n+w_n)_{n\geq 1}$
You surely have the right idea, but you might want to elaborate that if $z$ is a limit point of $(z_n)_n$ then $z_{n_k} \to z$ for some subsequence $(z_{n_k})_k$ of $(z_n)_n$. It follows that $(z_{n_k} + w_{n_k})_k$ is a subsequence of $(z_n + w_n)_n$ with $$ z_{n_k} + w_{n_k} \to z + 0 = z $$ so that $z$ is a limit point of $(z_n + w_n)_n$.
For the opposite direction "$\Leftarrow$" assume that a limit point of $(z_n + w_n)_n$. Applying the $\Rightarrow$ part to $\tilde z_n = z_n + w_n$ and $\tilde w_n = -w_n$ implies that $z$ is a limit point of $$ \tilde z_n + \tilde w_n = (z_n + w_n) + (-w_n) = z_n \, . $$
you can apply the same argument to $z_n = (z_n + w_n) - w_n$.