To show: Sequence $\langle u_n\rangle$ is convergent and converges to zero,where $u_n=\frac1{1n} + \frac1{2(n-1)} +\frac1{3(n-2)}+\dots+\frac1{n1}$

To show: Sequence $\langle u_n\rangle$ is convergent and converges to zero,where $u_n=\frac{1}{1n} + \frac{1}{2(n-1)} +\frac{1}{3(n-2)}+ \dots +\frac{1}{n1}$

Using Rabee's test I have shown that the sequence $\langle u_n\rangle$ is convergent.
Now to find the limit:$\frac{1}{nn} + \frac{1}{nn} +\frac{1}{nn}+ \dots +\frac{1}{nn}\le \frac{1}{1n} + \frac{1}{2(n-1)} +\frac{1}{3(n-2)}+ \dots +\frac{1}{n1}$
$\implies\frac{1}{n^2} + \frac{1}{n^2} +\frac{1}{n^2}+ \dots +\frac{1}{n^2}\le \frac{1}{1n} + \frac{1}{2(n-1)} +\frac{1}{3(n-2)}+ \dots +\frac{1}{n1}$
$\implies\frac{1}{n}\le \frac{1}{1n} + \frac{1}{2(n-1)} +\frac{1}{3(n-2)}+ \dots +\frac{1}{n1}$
I am unable to find an upper bound,which converges to zero.
Kindly help me. Also,if there are different methods then kindly share.