does this sequence necessarily converge? Let $\{x_n\}$ be a real sequence that satisfies $|x_{n+1} - x_n| < \frac{1}{n}$ for all $n \geq 1$.
Suppose we know that $\{x_n\}$ is bounded, then must $\{x_n\}$ converge?

No.  For example, starting at $x_1 = 0$ let $x_{n+1} = x_n + 1/n$ until $x_n > 1$, then $x_{n+1} = x_n - 1/n$ until $x_n < 0$, then $+$ again ...  Because $\sum_n 1/n$ diverges, you will have infinitely many terms $> 1$ and infinitely many $< 0$.
