Let $\pi:\mathbb{N}\to\mathbb{N}$ be a bijection. Then does there exist another bijection $\nu:\mathbb{N}\to\mathbb{N}$ such that $$ \frac{1}{n} + \frac{1}{\pi(n)} \leq \frac{2}{\nu(n)} \quad ? $$ If not, does it hold with $2$ replaced by a suitably larger constant?

While the harmonic sequence $(\frac{1}{n})_{n\in\mathbb{N}}$ is what comes up in my application, I imagine that a good answer will be able to make a much more general statement about a suitable class of sequences. But I'd be perfectly happy with an answer to the question above.