On sequentially closed subsets of the real line in ZF

(i) CAC iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. (ii) Every infinite subset X of R has a countably infinite subset iff every infinite sequentially closed subset of R includes an infinite closed subset. (iii) The statement “R is sequential” is equivalent to each one of the following propositions: (a) Every sequentially closed subset A of R includes a countable cofinal subset C, (b) for every sequentially closed subset A of R, A\A is a meager subset of A, (c) for every sequentially closed subset A of R, A\A 6= A, (d) every sequentially closed subset of R is separable, (e) every sequentially closed subset of R is Cantor complete, (f) every complete subspace of R is Cantor complete.