Suppose $L \subseteq \mathbb N$ such that, for the purpose of Turing machine computation, numbers in $L$ are represented by strings over the alphabet $\{0, 1\}$ in the standard binary notation. Prove that $L$ is decidable if and only if $L$ is finite or $L$ is the image of some strictly increasing computable function $f : \mathbb N \to \mathbb N$. Such a function $f$ is strictly increasing if for all $n_1$, $n_2$ $∈$ $\mathbb N$, $n_2$ $>$ $n_1$ implies $f(n_2) > f(n_1)$.
My attempt for the backward direction:
$L$ decidable $\impliedby$ $L$ finite.
If $L$ is finite we can simply check whether an input is in the set of finite words in $L$.
$L$ decidable $\impliedby$ $L$ is the image of $f$.
If $L$ is the image of some strictly increasing computable function there's a 1-1 correspondence between the natural numbers and the words in $L$. Therefore consider a Turing machine where on input $x$, we start incrementing a counter $y$, which is initially at $0$ while $f(y)<x$.
Once $y$ has finished incrementing, if $f(y)$ equals $x$ we accept else we reject. Since the number $y$ must be finite, the Turing machine always terminates. Therefore $L$ is decidable.
How can I proceed with the forward direction?