Proving two sequences are not homeomorphic

Question

Find an example of two real and limited sequences $(a_n)$ and $(b_n)$ such that $\{a_n:n\in\mathbb{N}\}$ and $\{b_n:n\in\mathbb{N}\}$ that are not homeomorphic fot the subspace topology of $\mathbb{R}$.

The following example was given $U_n=\frac{1}{n}$ for all $n\geqslant 1$ and $V_n=\begin{cases} 0, & \mbox{ if }n=1\\ \frac{1}{n}, & \mbox{ if }n>1 \end{cases}$

Suppose there exists a homeomorphism $f:\{V_n,n\geqslant 0\}\to\{U_n:n\geqslant 0\}$

Question:

What is the reason behind the fact $f$ is not an homeomorphism?

How should it be proved?

Thanks in advance!

Answer 1

Hint: One element of $\{V_n\,|\,n\in\mathbb{N}\}$ is an accumulation of $\{V_n\,|\,n\in\mathbb{N}\}$, but no point of $\{U_n\,|\,n\in\mathbb{N}\}$ has the similar property.