Prove that the sequence $x_n = 1 +\frac{ sin (n+ \pi) }{n} $ is a cauchy sequence

Question

Prove that the sequence $x_n = 1 +\frac{ sin (n+ \pi) }{n} $ is a cauchy sequence using the definition:

$$\forall \epsilon>0 \exists N\in\mathbb{N}: n,m\ge N\implies |x_n-x_m|<\epsilon.$$

I have tried to prove: $ | \frac{ n-sin(n)}{n} – \frac{ m -sin(m)}{m} | \leq \epsilon$.
The triangle inequality did not work for me and I don't know how to prove it with the provided definition.

I could argue that the sequence converges to 1 and is therefore cauchy. However, I need to prove this with the definition.

Please help me.

Answer 1

$|\frac{\sin (π+m)}{m}-\frac{\sin (π+n)}{n}|\le$

$|\frac{\sin (π+m)}{m}| +|\frac{\sin (π+n)}{n}|\le$

$1/m+1/n;$

Let $\epsilon >0$ be given.

Choose $n_0 > 2/\epsilon$ (Archimedean principle).

For $m\ge n \ge n_0:$

$|\frac{\sin (π+m)}{m}-\frac{\sin (π+n)}{n}|\le 1/m+1/n \le 2/n \le 2/n_0 <\epsilon.$