Integration Inequality for $C^{1}$ functions

Let $f,g$ be $C^{1}(R)$ which are nonnegative. $f'(x) < g'(x) \hspace{3mm} \forall{x} \in (0,s) $, does that mean $$\int_{0}^{s} f'(x) < \int_{0}^{s} g'(x)$$

I know that $a_{n} < b_{n} \forall{n}$ implies that $$\lim_{n} a_{n} \leq \lim_{n}b_{n}$$Since Riemann integration involves taking limits, maybe the $<$ should be replaced with$\leq$. Can someone verify?