I don't entirely understand the Axiom of Dependent Choice, and I couldn't find any simple examples anywhere; so I made one: suppose there exists a nonempty relation $R$ on the set $S=\{1,2,3\}$ such that $R=\{(1,1),(2,1),(3,1)\}$ ($R$ relates every element of $S$ to 1). What is the sequence $(x_n)$ of elements within $S$ such that for all $k \in \mathbb{N}$, $x_k R x_{k+1}$?
2026-04-06 00:33:44.1775435624
Simple Finite Application of Dependent Choice
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How about $x_k=1,$ so $1R1,1R1,1R1,\dotsb$. Along with $2R1,1R1,1R1,\dotsb$ and $3R1,1R1,1R1,\dotsb,$ those are the only choices.
Note that I did not invoke the axiom of dependent choice to construct this sequence.