Question on Convergence of sequence

Question

I have trouble with this question and I'm not really sure how to solve it correctly.

Let $\{a_n\}_{n\in \mathbb{N}} \neq 0$ be a complex sequence with only one accumulation point, $a$, for all n. Show that $\{a_n\}$ converges to a if $0$ isn't an accumulation point for $\{\frac{1}{a_n}\}$.

There is a theorem saying that a bounded complex sequence with only one accumulation point is convergent, which I'm pretty sure that we have to use.

Thanks a lot!

Answer 1

There is a theorem saying that a bounded complex sequence with only one accumulation point is convergent, which I'm pretty sure that we have to use.

Well, then let us show that the sequence is indeed bounded: Assume it is not, then for each $k \in \mathbb N$ there exists a $n_k \in \mathbb N$ with $|a_{n_k}| \gt k$. But then we have $|\frac{1}{a_{n_k}}| \lt \frac{1}{k}$ for all $k \in \mathbb N$, hence $0$ would be an accumulation point of $\{\frac{1}{a_n}\}$.

Answer 2

Asserting that $0$ is not an accumulation point of $\left(\frac1{a_n}\right)_{n\in\mathbb N}$ is equivalent to asserting that the sequence is bounded. That, together with the theorem that you mentioned, solves your problem.

Answer 3

$0$ is not an accumulation point for $\{\frac{1}{a_n}\} \iff \lim_{n\rightarrow\infty}\frac{1}{a_n}\neq0 \iff \lim_{n\rightarrow\infty}{a_n}\neq\infty$, which says the sequence is bounded. Then you can use the theorem you mentioned.