Convergence or divergence

Assume the limit exists and is finite. Then $\begin{aligned} 0 &= \lim_{n\to\infty} x_{n+1} - \lim_{n\to\infty} x_n \\ &= \lim_{n\to\infty} x_{n+1} - x_n \\ &= \lim_{n\to\infty} \frac{1}{(x_n)!} \end{aligned}$ which in particular implies $\lim_{n\to\infty} \Big|(x_n)!\Big| = \infty$

However, since obviously $x_n \geq x_1 = 1,$ the only singularity it could be approaching is at infinity, giving a limit of $\lim_{n\to\infty} x_n = \infty,$ a contradiction to the assumption the limit is finite, making the statement in the problem to be $\boxed{\text{no}}.$

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Note : If the limit had come back as $2e,$ then we'd have to actually prove the limit exists to make our conclusion. However, since we show the limit is infinite, the assumption that the limit exists and is finite is harmless to conclude the statement is false.