How to show that $\lim_{n \to \infty} \sum_{i=n}^\infty x_i = 0$ Let $\{ x_i \}$ be a decreasing sequence of non-negative real numbers such that $\sum_i x_i &lt; \infty$.

How do I argue that 

\begin{align}
\lim_{n \to \infty} \sum_{i=n}^\infty x_i = 0
\end{align}
Hint:

Let $$S=\sum_{i=1}^\infty x_i$$

Then, $$\sum_{i=n}^\infty x_i = S - \sum_{i=1}^{n-1} x_i$$

Now, take the limits of both sides.
The statement seems very much false.

$$x_i&gt;0 \hspace{4pt} \forall \hspace{2pt} i$$
$$\Rightarrow \sum_{i=n}^\infty x_i&gt;0$$