While evaluating the limits of sequences and writing down their proofs, I almost never get full points, despite the "actual" proof being correct. The mistakes are solely concerning notation.
For example, consider the sequence $$(x_n)_{n \in \mathbb{N}}= (-1)^n+2$$
The task is to find all the accumulation points of the sequence and find all the accumulation points of the set $ \{x_n | n \in \mathbb{N}\}$.
Now despite this being painfully obvious, I would create two subsequences $x_{2n}$ and $x_{2n-1}$ and show that they converge to 3 and 1 respectively. Is there any need to show that $(-1)^n$ converges to 1 for even integers $ n \in \mathbb{N} $?
Secondly, I'd claim that if $\lim\limits_{n \rightarrow \infty }{x_{2n}}=3$ that I know that $ \{x_n | n \in \mathbb{N}\}$ has an accumulation point at 3. I'd do it in the same way for 1. This follows from the fact that for a certain $n_0$ every member of the sequence is in the epsilon neighborhood of the limit, which is 3. And that is the very definition of accumulation points of sets.
What other things need to be said in order for this to become a proper proof?
A sequence $(x_n)_{n\in \mathbf N}$ is the image of a function $x\colon \mathbf N \to \mathbf R$ (and hence a set) where you define $x(n)=:x_n$ for all $n\in \mathbf N$.
The notation $(x_n)_{n \in \mathbb{N}}= (-1)^n+2$ is not good even if I think that everybody will understand what you want.
Your ideas are okay. First consider the sequence $(x_n)_{n\in \mathbf N}$ with $x_n = (-1)^n +2$. $(x_n)_{n\in \mathbf N}$ is not convergent since we can pick the subsequences $(a_n)_{n\in \mathbf N}$ and $(b_n)_{n\in \mathbf N}$ with $a_n=x_{2n}$ and $b_{2n+1}$ and see/show that $a_n\to 3$ (since we have $a_n=3$ for all $n\in \mathbf N$) and $b_n \to 1$. Therefore $1$ and $3$ are limit points of $(x_n)_{n\in \mathbf N}$.
In my opinion you have finally to show that these are the only limit points. (This can be easily shown by contradiction.)