If $x_{n}$ is decreasing and $\sum x_{n}$ converges, prove that $\lim nx_{n} = 0$ [duplicate]
First, we have that $x_n \geq 0$ for all $n$. Now, consider a "Cauchy slice" with $n$ terms: $$ C_n\stackrel{\rm{}def}{=}\sum_{k=n+1}^{2n} x_k $$ What can you say about it? And by what can you lower bound it?
Hint
By index change prove: $$\sum_{k=1}^n x_k=\sum_{k=1}^n k(x_k-x_{k+1})+nx_{n+1}$$
then prove that the series $$\sum_{n=1}^\infty n(x_n-x_{n+1})$$ is convergent and deduce that the sequence $(nx_{n+1})$ is convergent to some $\lambda$.
Can $\lambda$ be different to zero? Why? and finaly how to prove that the sequence $(nx_n)$ converges also to zero?