Hi everyone I was wondering if anybody could help me with this question; I have done the first 2 parts and a seperate 4th part but part 3 is causing me mathematical pain.

Suppose that

satisfies

.

1) Find f(0).

[easy to show that f(0) = 0 ]
2) Show that

[can be done easily by induction]
Now suppose that f is differentiable at 0.

3) Show that f(t) = t f'(0) for all

[Hint: For fixed t, consider the sequence

OK, we know that f is differentiable at 0, so

and for a fixed t, the above sequence simplifies to

which is a constant sequence, but how does this help me? I guess I can say that

Any help would be appreciated :)