MathDB

We say that two sequences

x,y \colon \mathbb{N} \to \mathbb{N}

are completely different if

x_n \neq y_n

holds for all

n\in \mathbb{N}

. Let

F

be a function assigning a natural number to every sequence of natural numbers such that

F(x)\neq F(y)

for any pair of completely different sequences

x

y

, and for constant sequences we have

F \left((k,k,\dots)\right)=k

. Prove that there exists

n\in \mathbb{N}

such that

F(x)=x_{n}

for all sequences

x

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