State and prove Gronwall's lemma.
Lemma: Let be continuous for and satisfy the inequalities
for some non-negative constants and . Then
It is sufficient to consider the case . The proof for the other case is the same. Let
Then is continuous and differentiable, and . Hence the inequality takes the form
Now multiply both sides by the non-negative function , so that
Now integrate the inequality from to :
which completes the proof.