EDIT: I meant to have the coefficients reversed, showing: $$\frac{n}{n-1}(1-(1-x)^n)^n + (1-x)^{n-1} \leq 1$$ This version should be true.. but still trying to prove it...
ORIGINAL:
Is it possible to show:
$$(1-(1-x)^n)^n + \frac{n}{n-1}(1-x)^{n-1} \leq 1$$ for $0 the second term seems to decrease with $n$, so I can solve for the minimum of $n$. But the first term doesn't completely increase with $n$ -- the lines cross when I plot the first term with $n=2$ and then $n=3$. So I'm not sure where to go from there.