Question

5. Let X be a metric space. (a) Let x E X be an isolated point. Prove that the only sequences in X that converge to a are the sequences that are eventually constant (b) Prove that the only convergent sequences in a discrete metric space (See Problem 8 on page 79 for the definitions of "isolated" and "discrete.") with tail a,z,x.... are the eventually constant sequences.

Answer 1

We use the definition of isolated point, and discrete metric space properly. First we prove (a), and then we use (a) to prove (b). By $B(x,\delta),$ or alternatively $B_{\delta}(x)$, we mean the ball of radius $\delta$ around $x$.