Prove $ \bigcap_{n=1}^{\infty} I_{n}=\bigcap_{n=1}^{\infty}\left[-1,1-\frac{1}{n}\right]=[-1,0] $

Question

From left to right

$x \in \bigcap_{n=1}^{\infty}\left[-1,1-\frac{1}{n}\right] \quad \Longrightarrow \quad x \in[-1,0]$

Proof by contraposition. I have to prove the following implication

$x \notin[-1,0] \Longrightarrow x \notin \bigcap_{n=1}^{\infty}\left[-1,1-\frac{1}{n}\right]$

Case 1: If $x < -1$, then $x$ is not a member of $I_n$. This is so trivial that I do not know how to proof it.

Case 2: $0<x$. I do not have any ideas. Been working on this for two days.

Thanks in advance