“Elementary properties of closures”

From Terence Tao’s Analysis I, p213 of the hardcover second edition, Lemma 9.1.11.

Showing $X\subset \overline{X}$. Let $x\in X$, and let $\epsilon >0$. We have $\left|{x-x}\right| = 0 \leq \epsilon$, so we are done.