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.

#Maximum #Sets #Decimals

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Maximums of the Set \(A\)

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