How do I show $y\in B(x,\delta)\implies B(y, \delta-||y-x||_{\infty})\subseteq B(x,\delta)$?

This is from "Supplement to Measure, Integration & Real Analysis" by Sheldon Axler. The definition of $B(x,\delta)$ is reproduced below.

For $x\in\mathbb{R}^n$ and $\delta>0$, the open cube$B(x,\delta)$ is defined by $B(x,\delta)=\{y\in\mathbb{R}^n:||y-x||_\infty<\delta\}$.

I can convince myself of this implication intuitively by taking $n=1$ but am unsure how to formalise this. I know it's elementary but I've been stuck on it and would appreciate some help. Thanks for considering my question!