As the title states, I would like to know if the statement $$ \sum_{j=1}^{\infty} x_{j}<\infty \Longrightarrow \lim _{k \to \infty} \sum_{j=k}^{\infty} x_{j}=0,\qquad x_j\in [0,\infty) $$ is always true?
Edit:
My confusion came from the following proof of convergence of the series $$ \sum_{j=1}^{\infty}\left\langle x, e_{j}\right\rangle e_{j} $$ (we are working in a Hilbert space, and the vectors $\{e_j\}$ are orthonormal.) Let $m,k\in \mathbb{N}$. Then $$ \begin{gathered} \left\|\sum_{j=1}^{m+k}\left\langle x, e_{j}\right\rangle e_{j}-\sum_{j=1}^{m}\left\langle x, e_{j}\right\rangle e_{j}\right\|^{2}=\left\|\sum_{j=m+1}^{m+k}\left\langle x, e_{j}\right\rangle e_{j}\right\|^{2} \\ =\sum_{j=m+1}^{m+k}\left|\left\langle x, e_{j}\right\rangle\right|^{2} \leq \sum_{j=m+1}^{\infty}\left|\left\langle x, e_{j}\right\rangle\right|^{2} \end{gathered} $$ We know that $\sum_{j=1}^{\infty}\left|\left\langle x, e_{j}\right\rangle\right|^{2}<\infty$ and it follows that $$ \lim _{m \rightarrow \infty} \sum_{j=m+1}^{\infty}\left|\left\langle x, e_{j}\right\rangle\right|^{2}=0 $$
Yes, it is true.
Let $S_n=\sum_{j=1}^n x_j,~x_j\ge0$, since $S_n$ is non-decreasing and bounded above, the partial sum is convergent, $\lim_{n\to\infty} S_n=S$, hence, the series is convergent.
$$S=\sum_{j=1}^n x_j+\sum_{j=n}^\infty x_j=S_n+T_n\Rightarrow T_n=S-S_n$$
Since $T_n$ is non-increasing and bounded below $T_n\ge0$, hence, $T_n$ is convergent.
$$\lim_{n\to\infty} T_n=\lim_{n\to\infty}S-\lim_{n\to\infty} S_n=S-S=0$$