Question about Theorem 6.27 in Rudin PMA on rectifiable curves

Question

I have a question about the proof for theorem 6.27 in rudin's principles of mathematical analysis where he argues $\Lambda(\gamma) \leq \int_a^b{|\gamma'(t)|dt}$. He argues that since $\gamma'$ is uniformly continuous (for every $\epsilon > 0$, $\exists \delta > 0$ such that $|s-t| < \delta \rightarrow |\gamma'(s)- \gamma'(t)| < \epsilon$), that given a partition $P$ such that $\triangle x_i < \delta $ and $x_{i-1}\leq t \leq x_i$, then $|\gamma'(t)| \leq |\gamma'(x_i)| + \epsilon$. Why isnt this inequality strict since $|t-x_i| < \delta$ is? Thanks.