I need to confirm that i am approaching this problem correctly.
Problem: Let $f$ and $g$ be functions $f,g:\mathbb{R}\to \mathbb{R}$
Prove that if $f^2 + g^2$ is bounded then $f$ is bounded.
Attempted Solution:
Definition of bounded: $|f(x)| \leq M$ for all $x \in \mathbb{R}$
Using that definition we can say
$|f^2 + g^2| \leq M$
Then: $|f| = \sqrt{|f^2|} \leq \sqrt{M}$
Therefore we can conclude that $|f|$ is bounded when $|f^2 + g^2|$ is bounded.
Is this true? Or have I gone horrible wrong? Any help is appreciated. Oh and if someone can point me to a good place to learn latex commands that would be sweet, thanks!