Alternate Form of Estimation Lemma for Product of Two Functions

Question

I am curious if this is a valid inequality:

$|\int_Xfg dx| \leq M|\int_Xgdx|$

Where $f,g$ are integrable functions and $M=\sup_X|f|$. Here was my (quick) proof:

$|\int_X fg dx| \leq \int_X |fg| dx \leq \int_X|f||g|dx \leq M\int_X|g|dx$

Is this valid? And if it is, would it hold if $X=\mathbb{R}$?

Answer 1

Hint: Consider $f(x) = g(x) = x$ on $[-1, 1]$.