Your proof relies on the Calculus of Taylor Polynomials. While this is perfectly valid, I'm not sure this method is allowed in the class that posed the problem. If you were to use calculus, you'd have to:
1) Find the T.P. of e^x (which is 1/0! + 1/1! + 1/2! ... + 1/n!)
2) Use the error formula and its approximations, and knowledge that e < 3 (if you can find that out) to find a reasonable way to squeeze e between two inequalities.
3) You'd pick an n to which you'd take the T.P. and multiply the inequalities through by n!

Then it will follow that n!(e - T.P.(e)) is an integer, but it is greater than 0 and less than 3/4, which is impossible. Thus e cannot be rational.

I think some of the earlier replies are actually good, if carefully checked.