How to show if $\sum_{n=1}^\infty 7X_{n+1} + X_n$ converges and $X_n \to 0$ when $n \to \infty$, then $\sum_{n=1}^\infty X_n$ converges? Note that $X_n$ is just a sequence of real numbers.

I'm a bit out of ideas because I'm used to having the extra hypothesis of $X_n$ being of positive terms. Any hints or suggestions will be greatly appreciated!
Let $L:=\sum_{n=1}^\infty (7X_{n+1} + X_n)$ and $S_N=\sum_{n=1}^N X_n$. Note that $$\sum_{n=1}^N( 7X_{n+1} + X_n) = 8S_N + 7X_{N+1}-7X_1$$
thus $\lim_{N\to \infty} (8S_N + 7X_{N+1}-7X_1) = L$. From the hypothesis $\lim_N X_N=0$, we can infer that $S_N$ converges and $\lim_{N\to \infty} S_N = \frac 18(7X_1+L)$.
$$\sum _{n=1}^\infty 7X_{n+1}+X_n=X_1+8\sum _{n=2}^\infty X_n$$
The second hypothesis is unneccessary. Indeed, it is implied by the first.