$\frac{da}{dt} \leq f(a(t))$ and $\frac{db}{dt}=f(b(t))$ imply $a(t) \leq b(t)$? Let $a(t)$ and $b(t)$ be nonnegative real-valued differentiable functions and $f$ be a strictly positive continuous function.
If $a(0)=b(0)$, $\frac{da}{dt} \leq f(a(t))$ and $\frac{db}{dt}=f(b(t))$ for $t \geq 0$. Then how do I show that $a(t) \leq b(t)$ for $t \geq 0$?
It seems intuitively clear but very tricky to prove....I cannot compare $f(a(t))$ and $f(b(t))$ directly. Could anyone help me?

See 
https://users.math.msu.edu/users/sen/Math_848/lec_6s.pdf
Inequality of differential equation implies inequality of functions.
