# $u_n=\frac {1}{1\cdot n}+\frac {1}{2(n-1)}+\ldots+\frac {1} {n \cdot 1}$,then $\lim u_n=?$

I am stuck on the following problem and do not know to tackle it:

If $u_n=\dfrac {1}{1\cdot n}+\dfrac {1}{2 \cdot (n-1)}+\dfrac {1}{3 \cdot (n-2)} + \ldots +\dfrac {1}{n\cdot1}$, then $\lim u_n = 0$.

Can someone point me in the right direction?

Hint: $(n+1)u_n=(1+\frac {1}{n})+(\frac {1}{2} +\frac {1}{n-1})+....+(\frac {1}{n}+1)$
First show that $u_n$ is bounded above by $1$ and below by $0$. Both are fairly straightforward. To show the upper bound, note that each term in $u_n$ is $\leq \dfrac1n$ (Why?).