Problem 100 (difficulty: 8/10)

Let \(\displaystyle A_1,A_2,\ldots\) be non-empty finite sets, and for all positive integer \(\displaystyle n\) let \(\displaystyle f_n\) be a map from \(\displaystyle A_{n+1}\) to \(\displaystyle A_n\)Prove that there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\) sorozat, such that for all \(\displaystyle n\) the conditions \(\displaystyle x_n\in A_n\) and \(\displaystyle f_n(x_{n+_1})=x_n\) hold. (König's lemma)

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