Find the limit of a recursively defined sequence

Question

A sequence is recursively defined sequence.

$u_{n+1} = \frac{n}{n+1} u_{n}$, where $u_{1} = 1$

From the monotonic bounded theorem, one can show that a limit for this sequence exists. How to compute the limit?

Answer 1

Note that $(n+1)u_{n+1}=nu_n$ so that the sequence $(n u_n)_n$ is constant, thus equal to its first term $u_1$.

Therefore $u_n=\frac{u_1}{n}$ and $\lim_n u_n = 0$.