real analysis - Show the equivalence $x=1$ $⟺$ $-ε

Let $x\in \mathbb{R}$. Show the equivalence $x=1 \Leftrightarrow-\epsilon\lt (x-1)\lt \epsilon$ for all $ε\gt0$.

So the first thing I thought to do was to prove both sides ($\Leftarrow$ and $\Rightarrow$) since this is an equivalence question.

i) To show $\Leftarrow$:

Suppose $x=1$, we have to show $-\epsilon \lt (x-1)\lt \epsilon$ for all $\epsilon\gt0$.

$1
If $x=1$ then $1\lt (1+\epsilon )$, $\epsilon \gt0$

$(x-1)\lt \epsilon$
$(x-\epsilon )\lt 1$

If $x=1$ then $1-ε<1$

Therefore $-\epsilon \lt 0$, so $\epsilon \gt 0$.

Then i would go on to prove the $\Rightarrow$side, but i'm not sure if i'm on the right line or not. Would really appreciate the help.