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I have a few uncertainties on this particular proof: http://planetmath.org/proofofgronwallslemma

Why are we allowed to divide at all in the first step, I think nothing prevents the right hand side form being zero, and assuming neither component is zero it is not trivial from my perspective. Does it make sense to add some $\epsilon > 0 $ before dividing? Does this hold up later on in the proof?

Secondly I do not understand the step where they integrate at all, what happens with the numerator?

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    I don't know the website, but I find it weird to not include conditions on the functions. Nevertheless, in Grönwall lemma, as stated on the website you provided, we have $K,L \geq 0$, $t_0 \leq t$ and $\psi, \phi \geq 0$. So the right hand side is always positive. In the particular case where it is zero, the proof is done. Otherwise, they just divide by a strictly positive value, which is fine. – 2017-01-15
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    Ah I missed that, I thought only $L \geq 0$, yes it's just a proof I found and liked. Do you know about the integration he performs? – 2017-01-15
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    Looks like a regular integration to me. Except the poor choice of name for the free variable as they change $s$ to $\tau$ before integration with respect to $t$. – 2017-01-15
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    Oh, it makes sense now, yes I was a bit confused by the variable changes. Thanks! – 2017-01-15

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