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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!

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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 $\lim_{N\to \infty} S_N = \frac 18(7X_1+L)$.

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Consider $S_k := \sum_{n=1}^k x_n$. Then your condition reads: \begin{align} (i) \quad & 7(S_{k+1}-S_1) + S_k \to L \quad \text{as } k \to \infty; \\ (ii) \quad & S_{k+1}-S_k \to 0 \quad \text{as } k \to \infty. \end{align} Note that $(i)$ is equivalent to saying that $7S_{k+1} + S_k$ converges, since $S_1$ is a fixed real number.

Now note that $7S_{k+1}+S_k = 8S_{k+1}-(S_{k+1}-S_k)$ converges. Can you use conition $(ii)$ to deduce the desired result?

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$$\sum _{n=1}^\infty 7X_{n+1}+X_n=X_1+8\sum _{n=2}^\infty X_n$$ The second hypothesis is unneccessary. Indeed, it is implied by the first.

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