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## Homework Statement

If [itex]n[/itex] is a positive integer and if [itex]x > 0[/itex], show that

[itex]\displaystyle\left(1 + \frac{x}{n}\right)^n < e^x[/itex] and that [itex]\displaystyle e^x < \left(1 - \frac{x}{n}\right)^{-n}[/itex] if [itex]\displaystyle x < n[/itex].

## The Attempt at a Solution

I have proved the first inequality, but I am confused about the second one. Although I know

[itex]\displaystyle \left(1 - \frac{x}{n}\right)^{-n} = \left(1 + \frac{x}{n-x}\right)^{n}[/itex],

but I have no idea for the next steps.