I'm reading a proof about the characterization of jump continuous functions.
$\textbf{Theorem}$ Let $I=[\alpha,\beta] \subseteq \mathbb R$ and $E$ a Banach space. Then $f:I \to E$ is jump continuous if and only if there is a sequence of staircase functions converging uniformly to $f$.
It seems to me that we only get $\left\|f-f_{n}\right\|_{\infty} \le 1 / n$ after taking the suppremum.
Could you please confirm if this strict inequality is a typo? Thank you so much!