Cardinality of set of all monotone increasing functions between $\Bbb N$

Question

Let $X =\{f:\Bbb N \rightarrow \Bbb N\mid f \text{ is monotone increasing function}\}$

Then want to find the $card(X)$

First what I know is function $f$ is equal to the monotone increasing sequence that is defined on $\Bbb N$

and I want to find any set which can be mapped from X to the set which would be injective and also another set which can be mapped from that set to X which would be injective.

Any suggestion for those kind of sets?

Answer 1

Think about the functions that have this property: $$f(n)\in\{2n-1,2n\}$$ for all $n$. Are these automatically increasing? How many of them are there?

Answer 2

Do you know how many functions there are $\Bbb N \to \Bbb N$? For any $f$ in your set, we can have $g(1)=f(1), g(n)=f(n)-f(n-1)$ This is a bijection between the $f$s and $g$s and I claim the $g$s are all functions $\Bbb N \to \Bbb N$. The bijection also says the set of $g$s could be your $X$.