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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!

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    I have removed the proof tag and replaced with algebra precalculus.2011-02-16

1 Answers 1

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There is an easier way, given that squares of real numbers are non-negative, so $f^2 \ge 0$, $g^2 \ge 0$ and $f^2 +g^2 \ge 0$.

If $ f^2 + g^2 \le M$ then $ f^2 \le M$, so $-\sqrt{M} \le f \le \sqrt{M}$.

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    Wow thats it? It makes sense and everything but....aghh. Thanks !2011-02-16