Exponential series/R/Estimate for remainder/Exercise

For $N\in \mathbb{N}$ and $x\in ℝ$ let

$R_{N+1}(x)=\exp x-\sum _{n=0}^N\frac{x^n}{n!}=\sum _{n=N+1}^{\mathrm{\infty }}\frac{x^n}{n!}$

be the remainder of the exponential series. Prove that for

$|x|\le 1+\frac{1}{2}N$

the remainder term estimate

$|R_{N+1}(x)|\le \frac{2}{(N+1)!}|x{|}^{N+1}$

holds.

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