Confusion on inequality in proof
We have $|x-y|<\delta$.
$$|y|=|y-x+x|\leq |x-y|+|x| <\delta +|x|$$
That is because the possible value of $y$ will be in $(x-\delta,x+\delta)$ since $|x-y|<\delta$.
Let $|x - y| < \delta$. Then, $$ x - \delta <y<x + \delta\\ |y| < \max\{| x - \delta|,|x+\delta|\} \le |x| + \delta $$ Therefore, $$ |y| - |x| < \delta $$