Spoken roughly, once it get's close, it never wanders off again. You must demonstrate, since you seem confident this converges to f(x) = 0 on the interval, that for some value of n, it is very close to zero on the entire interval AND that it never gets any worse than that.
For your sequence, the maximum deviation is always at x = 2/n. Does that do anything for us? Maximum deviation is 4/[(e^2)*(n^2)].