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. 

Hint: diagonalisation.
You'll probably want $f(0)=h_0(0)$ and then $f(1)=h_1(1)$ and $f(2)=h_0(2)$. Can you extend this using a diagonalisation argument? Can you then also show that $\{n \mid f(n)=h_m(n)\}$ is an infinite set?
