I wanted to examine the convergence of the series $\displaystyle x_{n+1}= \frac{n}{n+1} x_n$ and try to find its limit, but I'm having difficulty doing so. The only test I thought would be useful (ratio test) was inconclusive and I'm having trouble proving $0 \leq x_n \leq 1$ that I think could help end up with a geometric sequence. Any suggestions are welcome.
$x_1 = 1$ (This is $\forall n \geq 1$)