Tao, Analysis I, Definition of "$\epsilon$-closeness": Why weak inequality ($\leq$) instead of strict inequality ($<$)?

Question

From Tao, Analysis I, p. 87, bottom:

Why does he put $d(y,x)\leq \epsilon$ instead of $d(y,x)< \epsilon$ (which I think is more usual in these contexts)? What does this move achieve?

Answer 1

The two ways of defining it are essentially the same. If $d(x,y)\leq\epsilon$ you can find that for any $\epsilon'>\epsilon$ we have $d(x,y)<\epsilon'$, and clearly $d(x,y)<\epsilon$ implies $d(x,y)\leq\epsilon$, so it is not a big matter, unless there is somewhere later he needs the fact "$\{y\in\mathbb{R}:y\text{ is } \epsilon\text{-close to }x\}$ is a closed set". Otherwise, at least to me, it is just a personal preference to write $\leq$ instead of $<$.