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# Maximums of the Set $$A$$

Let $$A$$ be the set of numbers in $$(0,1)$$ whose decimal representations consist of only $$0$$s and (finitely many) $$1$$s. For example, $$0.1$$ is in this set, as is $$0.0011101$$ and $$0.11001$$.

Note that not every subset of $$A$$ has a maximum. For example, $$B=\{0.1,0.11,0.111,0.1111,\dots\}$$ has no maximum. (It has a limit $$0.\overline1=\frac19$$, but that's not in $$B$$ so it doesn't count.)

We can divide $$A$$ into pieces. Let $$A_1$$ be the set of numbers in $$A$$ with only one $$1$$. That is, $$A_1=\{\dots,0.001,0.01,0.1\}$$. Let $$A_2$$ be the set of numbers in $$A$$ with exactly two $$1$$s. That is, $$A_2=\{\dots,0.0101,0.011,\dots,0.101,0.11\}$$. More generally, let $$A_n$$ be the set of numbers in $$A$$ with exactly $$n$$ $$1$$s.

Note that $$A=A_1\cup A_2\cup A_3\cup\dotsb$$.

Prove that, for every $$n$$, every subset of $$A_n$$ has a maximum.

Note by Akiva Weinberger
2 years, 2 months ago