Is it true that if $(a_{n+1}-a_n)\to 0$ and $(a_n)$ is bounded, then $(a_n)$ is convergent

No: consider the sequence $$1,0,\frac12,1,\frac23,\frac 13,0,\frac 14,\frac24,\frac 34,1,\frac 45,\frac35,\frac25,\frac15,0,\ldots$$

So we know that it is false when we remove "bounded", just by letting $a_{n+1} - a_n = \frac1n$. This sequence "walks slowly to infinity"

We can produce a bounded example, just by setting up boundaries at say $-2$ and $2$ and reflecting the sequence as it progresses, so that it always stays within $[-2,2]$ but we still have $|a_{n+1}-a_n| = \frac1n$. Then this sequence doesn't converge.