Find the cardinality of $A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}$

Question

How can I solve tasks like this one?

Example task

Find the cardinality of $$A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}.$$

I know that $|A| \le \mathfrak{c}$ but how to deal with lower bound?

Answer 1

Notice that the set $\{f:\Bbb N\to\{0,1\}\mid f(0)=0\}=B⊆A$, so $|A|≥|B|=2^{|\Bbb N|}=\frak c$