Solution: We recall that if a sequence of real numbers $\{ a_n \} $ converging to $a$, then the arithmetic mean sequence $\frac{a_1 + a_2 + \dots + a_n}{n}$ also converges to $a$. Consider the sequence \[ S_n = \frac{1}{n}\sum_{i=1}^{n} (x_{i+1} - x_i) = \frac{x_{n+1} - x_1}{n} \rightarrow x. \] Since $\frac{x_1}{n} \to 0$, then \[ x = \lim_{n \to \infty} \frac{x_{n+1} - x_1}{n} = \lim_{n \to \infty} \frac{x_{n+1}}{n} - \lim_{n \to \infty } \frac{x_1}{n} = \lim_{n \to \infty} \frac{x_{n+1} }{n} = \lim_{n \to \infty} \frac{x_n}{n}. \]