Metric spaces/Continuous mapping/Characterization/Fact

Let

$f\colon L\longrightarrow M,x\longmapsto f(x),$

denote a mapping between the metric spaces ${}L$ and ${}M$. Then the following statements are equivalent.

1. ${}f$ is continuous in every point ${}x\in L$.
2. For every point ${}x\in L$ and every ${}\epsilon >0$, there exists a ${}\delta >0$ such that ${}d(x,x')\leq \delta$ implies that ${}d(f(x),f(x'))\leq \epsilon$ holds.
3. For every point ${}x\in L$ and every convergent sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ in ${}L$ with ${}\lim _{n\rightarrow \infty }x_{n}=x$, also the image sequence ${}{\left(f(x_{n})\right)}_{n\in \mathbb {N} }$ converges to the limit ${}f(x)$.
4. For every open set ${}V\subseteq M$, also the preimage ${}f^{-1}(V)$ is open.

Write a proof