Nested Quantifiers Proof

Question

Trying to wrap my head around how to prove something like this:

$$\exists x \in\Bbb R~, \forall y \in\Bbb R:(y \geqslant x) \lor [y \in (-\infty, -1) \cup (1, +\infty)]$$

Answer 1

Is true.

Take $x = -1$

Then if $y ≥ x = -1$ we are done. Else $y<x = -1 \implies y \in (-\infty, -1)$

So the condition its holds.