I am helping out a friend who can't seem to get these proofs and he enlisted my help thinking I know this. Unfortunately, I have tried wrapping my head around it and I can't find a solution to these problems.
Can someone tell me how to solve this or point me in the right direction with resources? Thanks!
Prove that for all real numbers x, y, and z, if x + y + z greater than or equal to 3, then either x greater than or equal to 1 or y greater than or equal to 1 or z greater than or equal to 1.
$$\forall x,y,z\in \mathbb R, \quad x+y+z \geq 3 \implies x \geq 1 \lor y \geq 1 \lor z\geq1 $$
Prove that for all real numbers x and y, if xy less than or equal to 2, then either x less than or equal to square root of 2 or y less than or equal to square root of 2.
$$\forall x,y \in \mathbb R,\quad xy \leq 2 \implies x \leq \sqrt 2 \lor y \leq \sqrt 2$$