Existence of a function,$f:\mathbb{N}\to \mathbb{N}$ I'm totally lost at this problem. Could anyone advise on the approach to it? 
Suppose $H: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is a function. For each $m\in \mathbb{N},$ define a function $h_{m}: \mathbb{N} \to \mathbb{N}$ as follows: $\forall n\in \mathbb{N} [h_{m}(n)=H(m,n)]$. Prove there is a function $f:\mathbb{N}\to \mathbb{N}$ such that for each $m\in \mathbb{N},\left\{n\in \mathbb{N}: f(n) = h_{m}(n)\right\}$ is an infinite set. 

Let $f\left(2^{k-1}(2l-1)\right):=h_k\left(2^{k-1}(2l-1)\right)$ for $k, l \in \mathbb{N}$.
