Let $ a_n $ be the $n $-th term of an infinite strictly increasing subsequence of $ \mathbb{N}$ and denote with $\nu(x)$ the number of terms smaller than or equal to $x$. Assume also $$f(x)<\nu(x)<g(x).$$ How can we find $f_1$ and $g_1$ such that $$f_1(n)<a_n<g_1(n) \ ?\tag{$\star$}$$ Obviously one could try replacing $x$ with $a_n$ in my first display, but what if that doesn't result in $(\star)$?
I'm not sure about the tags, please feel free to improve them if you think it is necessary.
Since $a_n = \min \{x : \nu(x) \geq n\}$, we have that if $f(x) > n$, then $x > a_n$ and if $g(x) < n$, then $x < a_n$. So we may define the somewhat artificial functions $f_1(x) = \sup \{x : g(x) < n\}$ and $g_1(x) = \inf \{x: f(x) > n\}$ which satisfy $f_1(x) \leq a_n \leq g_1(x)$.
If $f$ and $g$ are strictly increasing continuous functions, we can write those expressions more simply as $f_1(x) = g^{-1}(n)$ and $f_2(x) = f^{-1}(n)$.