Solution
= Solution
\b[No.] Take \b[$\boxed{a_n=\log n}$]. For each fixed positive integer $k$,
$$
a_{n+k}-a_n=\log(1+k/n)\longrightarrow0
$$
by continuity of the logarithm at $1$. Yet $a_n\to+\infty$, so $(a_n)$ is not a <convergent sequence> of real numbers. Vanishing differences at every fixed lag do not control the accumulation of many small increments.