I am trying to understand the proof to Gronwall's Inequality, but I have one major issue I was hoping someone could get me through. My issue actually makes the proof incorrect.

First, the statement of Gronwall's Inequality

Let

be continuous functions and let be a constant. Then if

For , it follows that

for .

Proof:

How do we know ?

How do we know ?

EDIT: I guess both my questions boil down to how do we know ? Since, we know by

EDIT2: . However, I still don't see how we know .