I want to show that $f+g \in L^p$ when both $f,g \in L^p$. The proof I'm give just says
Indeed $$|f(x) + g(x)|^p \le 2^p (|f(x)|^p + |g(x)|^p)$$ as can be seen by considering separately the cases $|f(x)| \le |g(x)| $ and $|g(x)| \le |f(x)|$.
Can anyone help me understand where $2^p$ came from and how the inequality holds?