if |f| + |g| are bounded are f and g bounded individually?

Question

I've tried this... but is this legit?

my sol:

Assume $|f| + |g|$ are bounded, then we have that $|(|f|+|g|)(x)| \leq M \quad \forall x \in \mathbb R$.

We have that $M \geq |(|f|+|g|)(x)| =||f(x)|+|g(x)||=|f(x)|+|g(x)| \geq |f(x)| \geq 0$

so $M \geq |f(x)|$. This same logic applies for $g(x)$. So there is a number $M$ that is always greater than $|f(x)|$ and $|g(x)| \quad \forall x \in \mathbb R$. So $f$ and $g$ are bounded.

Is this a logical explanation?

Thank you.

Answer 1

Yes, that's correct.

The symbol $\le$ means that the LHS is less than or equal to the RHS. It does not imply that equality is possible. In fact in this case the possibility of equality in $|f(x)| \le M$ is not the normal case. We have that $|f(x)|\le M-|g(x)|$ and for $|f(x)|$ to equal $M$ we would definitely need $|g(x)|=0$ which ought to be considered a special situation.