Axiom of dependent choice
In mathematics, the axiom of dependent choice is a weak form of the axiom of choice which is still sufficient to develop most of real analysis. The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (xn) in X such that xnRxn+1 for each n in N. (Here an entire binary relation on X is one such that for each a in X there is a b in X such that aRb.) Note that even without such an axiom we could form the first n terms of such a sequence, for any natural number n; the axiom of dependent choice merely says that we can form a whole sequence this way, which is intuitively obvious. See also: axiom of countable choice
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