I tried answering the following question, do my arguments make sense and are they correct?
Suppose we have a sequence $a_n$ with positive terms and accumulation points $0$ and $2$.
We consider the new sequence, $n\in \mathbb{N}_+$:
$$ b_n= \frac{n+a_n}{n \cdot a_n}$$
I am now asked to answer the questions:
(a) give an accumulation point of $b_n$
Well, notice that: $$ b_n= \frac{n+a_n}{n \cdot a_n}=\frac{1}{a_n} + \frac{1}{n}$$ In the limit for large $n$, we have that $\frac{1}{n} \rightarrow 0$. We notice that $a_n$ has a subsequence that converges to $2$, then certainly we have that $b_n$ has a subsequence $b_{n_j}$: $$ b_{n_j}=\frac{1}{a_{n_j}} + \frac{1}{{n_j}}\rightarrow \frac{1}{2}+0$$ So $\frac{1}{2}$ is an accumulation point.
(b) Is $b_n$ bounded?
Notice that we have subsequence $a_{n_k}$ that converges to $0$, we now consider $b_{n_k}$, such that: $$ b_{n_k}=\frac{1}{a_{n_k}} + \frac{1}{n_k}\rightarrow \infty $$ Because the sequence $b_n$ has a subsequence that is divergent to $\infty$, it cannot be bounded $\square$.
The argumentation seems to be correct.
I would just caution you to be a little more careful with notation.
How is $a'_n$ being labeled? Do the subindices coincide when passing to $b'_n$?