Extract from Lemma 4.21 M.Einsiedler & T.Ward.
$X$ is a normed space, $Y$ is Banach and $T:X \to Y$ is a bounded surjective linear map. The implication of the lemma shows that for any $\epsilon > 0$ there is $\delta >0$ such that $B_{\delta}^{Y} \subset \overline{TB_\epsilon^X}$.
... // Since $X=\cup_n nB_\epsilon^X$, $B_\epsilon^X=B_\epsilon^X(0)$, and $T$ is onto, we have $Y= \cup_n nTB_\epsilon^X$. By Baire's Theorem it follows that there is $n \geq 1$ such that the set $n\overline{TB_\epsilon^X}$ contains some ball $B_r^{Y}(z)$ in $Y$. Then, by linearity, $\overline{TB_\epsilon^X}$ must contain the ball $B_{r/n}^{Y}(z/n)$ //...
Why is the argument for linearity repeated in the last sentence, once we have obtained $B_r^{Y}(z) \subset n\overline{TB_\epsilon^X}$? Is it not the case that $\frac{1}{n}B_{r}^{Y}(z)=B_{r/n}^{Y}(z/n)$ and $V \subset nU \implies \frac{1}{n}V \subset U$ for sets $U,V$, so that $B_{r/n}^{Y}(z/n)=\frac{1}{n}B_{r}^{Y}(z) \subset \overline{TB_\epsilon^X}$? The purpose of writing $B_{n\epsilon}^{Y}=nB_{\epsilon}^{Y}$ I would have thought was to make the last line obvious from purely set-theoretical notions.