Extremely difficult induction proof problem

I dunno who thought up this one, but it is clearly for masochists.

Prove that for all $\displaystyle x > 0$ and all positive integers n

$\displaystyle e^x>1+x+\frac {x^2}{2!}+\frac {x^3}{3!}+...+\frac {x^n}{n!}$

$\displaystyle n!=n(n-1)(n-2)...3\cdot 2\cdot 1$

This MUST be proven by induction on $\displaystyle n\geq 1$.

In addition, these hints were provided for the question. They must be justified.

$\displaystyle e^x=1+\int_0^x e^t dt > 1+\int_0^x dt = 1+x$

$\displaystyle e^x=1+\int_0^x e^t dt > 1+\int_0^x (1+t) dt = 1+x+\frac {x^2}{2}$

As I said, whoever came up with this one is a sadist.

I believe I have correct justifications on the two hints (unless I wrote something wrong), but I can't seem to provide the proof for a $\displaystyle n+1$ case (for proving by induction). Still, if anyone can help explain this one in detail, I'd be very appreciative.