Real Analysis Show that if {x n } from n=1 to infinity is a convergent sequence in R d , then there exists N in the positive integers such that X N = X N+1 =X N+2 . . . (That is, a sequence in R d is...

Real Analysis

Show that if {x_n} from n=1 to infinity is a convergent sequence in R_d, then there exists N in the positive integers such that X_N
= X_N+1=X_N+2
. . .  (That is, a sequence in R_d
is convergent if and only if all the terms of the sequence are the same from some point on.)

Would you please be complete in your explanations of each portion of the answer?  I have never seen metric spaces before and am struggling to understand the concept.  Thank you.