Let (Xn) be a Cauchy sequence such that Xn is an integer for every n e N. Show that (Xn) is ultimately constant Solution If {x_n} is a Cauchy sequence, then, by definition, for any e > 0 you can find a positive integer N such that the distance between two members of the sequence x_n and x_m is less than e for all n,m > N. If e < 1, then the distance between x_n and x_m is less than 1 for n,m > N, where N is some positive integer. But x_n and x_m are integers, so they must be equal, because distance between distinct integers can\'t be less than 1, and the sequence is ultimately constant..

1. Let (Xn) be a Cauchy sequence such that Xn is an integer for every n e N. Show that (Xn) is
ultimately constant
Solution
If {x_n} is a Cauchy sequence, then, by definition, for any e > 0 you can find a
positive integer N such that the distance between two members of the sequence x_n and x_m is
less than e for all n,m > N. If e < 1, then the distance between x_n and x_m is less than 1 for n,m
> N, where N is some positive integer. But x_n and x_m are integers, so they must be equal,
because distance between distinct integers can't be less than 1, and the sequence is ultimately
constant.