$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$
Show that, $\lim_{n\rightarrow\infty} u_n = 0$.
The only approach I can see is either finding $nu_{n}$ or $(n+1)u_{n}$ and seeing that:
$$(n+1) \ u_{n+1} = (1+\frac{1}{n}) + (\frac{1}{2} + \frac{1}{n-1}) + \dots + (\frac{1}{n} + 1).$$
Please help me understand how I can solve this problem using this, or any method of your preference! Thank you so much!!