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Could you please help me with this question ?

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 $a_{n-1}<a_n, a_n>a_{n+1}$ and $f(a_{n-1})=f(a_n)=a_n$ ?

Thank you.

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

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  • 1
    $\begingroup$ It's almost like an "have you tried it on and off again" answer in IT $\endgroup$ – Mathematician 42 Oct 9 '18 at 6:27

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