- #1
Artusartos
- 247
- 0
Homework Statement
Let ##X## be a topological space. Let ##A_1 \supseteq A_2 \supseteq A_3...## be a sequence of closed subsets of ##X##. Suppose that ##a_i \in Ai## for all ##i## and that ##a_i \rightarrow b##. Prove that ##b \in \cap A_i##.
Homework Equations
The Attempt at a Solution
Suppose, for contradiction, that ##b \not\in \cap A_i##. Then there exists ##n \in \Bbb{N}## with ##b \not\in A_n##. But then ##b \not\in A_i## for all ##i \geq n##.
Since ##A_n## is closed, we can pick an open neighborhood ##V## of ##b \in A_1## with ##V \cap A_n = \emptyset##. By the definition of convergence, ##V## needs to contain all ##a_i## with ##i \geq N## for some ##N##. This is not possible because only a finite number of the ##a_i## can be contained ##V##, since ##b \not\in A_i## for all ##i \geq n##.
Is my answer correct?
Thanks in advance