Basic Analysis I — Real Analysis · Problem 7.3.11

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · sequences-and-convergence

Problem

Suppose

\{V_n\}_{n=1}^\infty

is a sequence of open sets in

(X,d)

such that

V_{n+1} \supset V_n

for all

n

. Let

\{x_n\}_{n=1}^\infty

be a sequence such that

x_n \in V_{n+1} \setminus V_n

and suppose

\{x_n\}_{n=1}^\infty

converges to

p \in X

. Show that

p \in \partial V

where

V = \bigcup_{n=1}^\infty V_n