Cauchy sequence. Using definition.

Question

Prove that

$$x_n = 1 + \frac{1}{2!} + \cdots + \frac{1}{n!}$$

is a Cauchy sequence.

I already showed that this sequence converges using

$$1 + \frac{1}{2!} + \cdots + \frac{1}{n!} < 1 + \frac{1}{2} + \cdots + \frac{1}{2^{n-1}}.$$

But I couldn't show that the is a Cauchy sequence using the definition. Any suggestion?

Answer 1

Given $\epsilon>0$, for large $N$, if $m>n\geq N$, $x_{m}-x_{n}=\dfrac{1}{(n+1)!}+\cdots+\dfrac{1}{m!}<\dfrac{1}{2^{n+1}}+\cdots+\dfrac{1}{2^{m}}<\epsilon$.