Let $\phi :[a,b] \rightarrow \Bbb R$ be a step function.
I have to prove that $|\phi|$ is a step function. Here's how I prove it:
Let $P$ be a partition $P=\{p_0,...,p_k\}$ on $[a,b]$, compatible with $\phi$. Let $\phi_i$ be the value that $\phi$ takes on each interval $(p_{i-1},p_i)$, for $1\leq i \leq k$. We see that $|\phi|$ takes values $|\phi_i|$ on the intervals $(p_i,p_{i-1})$, hence by definition, $|\phi|$ is a step function.
Is my proof correct? It seems too short to me, but I think there's nothing more to it really. Any feedback is appreciated