We’re going to need another analytic lemma, this one called “Gronwall’s inequality”. If is a continuous, nonnegative function, and if and are nonnegative constants such that
for all then for all in this interval we have
That is, we can conclude that grows no faster than an exponential function. Exponential growth may seem fast, but at least it doesn’t blow up to an infinite singularity in finite time, no matter what Kurzweil seems to think.
Anyway, first let’s deal with strictly positive . If we define
then by assumption we have . Differentiating, we find , and thus
Integrating, we find
Finally we can exponentiate to find
proving Gronwall’s inequality.
If , in our hypothesis, the hypothesis is true for any in its place, and so we see that for any positive , which means that must be zero, as required by Gronwall’s inequality in this case.