Here are a couple of others:
Let $H_n=\sum_{j=1}^n 1/j$. Then for all $n\geq 1$, $$ \sum_{d|n}d\leq H_n+(\log H_n)e^{H_n}. $$ Jeff Lagarias showed that this is equivalent to the Riemann hypothesis!
Let $x_0=2$, $x_{n+1}=x_n-\frac{1}{x_n}$ for $n\geq 1$. Then $x_n$ is unbounded.