TITLE: Given $\mu$ the counting measure on an infinite set $\Omega$, $\lim \mu(A_n) \ne 0$
QUESTION [5 upvotes]: Problem: Let $\mu$ be the counting measure on an infinite set $\Omega$.  Prove that there is a sequence of sets $A_1 \supset A_2 \supset A_3 \dots$ such that $\bigcap A_n = \varnothing$, but $\lim_{n \to \infty} \mu(A_n) \ne 0$.
Attempt:Choose a countably infinite set $\{x_i : i \in \mathbb{N} \}  = A_1 \subseteq \Omega$.   Define $A_n = \{x_i : i \ge n\}$  Then we have $A_1 \supseteq A_2 \dots$ with $\bigcap A_n = \varnothing$.
However, $\mu(A_n) = \infty$ for all $n$.
Does my proof look correct?

REPLY [3 votes]: Yes.${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$