# Is this function from $\mathbb{N}^\mathbb{N}$ injective?

Suppose $$f:\Bbb N^\Bbb N \to \Bbb N^\Bbb N$$ is function from set of sequences of natural numbers to set of sequences of natural numbers.

For each sequence $$(a_n|n \in \Bbb N):$$ $$f(a_n)=(b_n|n \in \Bbb N), b_n=max\{a_n,a_{n+1}\}$$.

Is $$f$$ injective ?

May I argue that if the set of sequences of natural numbers is fully ordered, then $$f$$ is always injective, otherwise, there exists some $$a_{n-1}a_{n+1}$$ and $$f(a_{n-1})=f(a_n)=a_n$$ ?

Thank you.

No. $$f(1,2,2,...)=f(2,2,,2...)$$ so $$f$$ is not injective.