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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

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    $\begingroup$ This proof is correct. $\endgroup$ – Boka Peer Mar 25 '20 at 13:24

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