Doubt in an epsilon delta proof step

Question

I am confused with a step in a $\epsilon$-$\delta$ proof. Let's consider $[a,b]$. Let's consider a partition of this interval the usual way with Riemann integrals. Let $f:[a,b]\to\mathbb{R}$ be a continuous function. Consider an that the following inequality holds $$ \bigg|\int_a^b\,|f(x)|\,dx-\sum_{k=1}^n|f(x)|(t_k-t_{k-1})\bigg|<\epsilon $$ using the triangle inequality $|a|-|b|\leq|a-b|$ we can write $$ \int_a^b\,|f(x)|\,dx-\sum_{k=1}^n|f(x)|(t_k-t_{k-1})\leq\bigg|\int_a^b\,|f(x)|\,dx-\sum_{k=1}^n|f(x)|(t_k-t_{k-1})\bigg|<\epsilon $$ this implies $$ \int_a^b\,|f(x)|\,dx<\sum_{k=1}^n|f(x)|(t_k-t_{k-1})+\epsilon $$ nonetheless my my book writes $\leq$ instead of $<$ $$ \int_a^b\,|f(x)|\,dx\leq\sum_{k=1}^n|f(x)|(t_k-t_{k-1})+\epsilon $$ why not the strict inequality?