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.

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