- #1
divergentgrad
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Homework Statement
Suppose [itex]\Omega[/itex] is an infinite set. If [itex] Q = \{x_1,x_2,...\} \subset \Omega [/itex] is infinite and countable, and if [itex] B_n := \{x_1,x_2,...,x_n\}, A_n := Q - B_n [/itex], ...
does [itex]A_n \downarrow \emptyset[/itex]? If [itex]\mu[/itex] is the counting measure on [itex]\Omega[/itex], is [itex] \lim_{n \to \infty} \mu (A_n) = 0[/itex]?
The Attempt at a Solution
My first thought is that [tex] \lim_{n \to \infty} \lim_{m \to \infty} (m - n) = \infty [/tex] would neatly parallel the above situation, and suggest that [itex]A_n [/itex] does not approach [itex]\emptyset[/itex]. Is that correct?
Now I'm actually doubting myself. Is it true that [tex] \lim_{n \to \infty} \mbox{ } \lim_{m \to \infty} (m - n) = \infty \neq -\infty = \lim_{m \to \infty}\mbox{ } \lim_{n \to \infty} (m - n) [/tex]? Or are these just indeterminate?
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