Exercise 1.6.4

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%1.6.4
  Let $S$ be the set consisting of all sequences of 0 's and 1 's. Observe that $S$ is not a particular sequence, but rather a large set whose elements are sequences; namely,
  $$
  S=\left\{\left(a_{1}, a_{2}, a_{3}, \ldotsight): a_{n}=0 	ext { or } 1ight\}
  $$
  As an example, the sequence $(1,0,1,0,1,0,1,0, \ldots)$ is an element of $S$, as is the sequence $(1,1,1,1,1,1, \ldots)$.
  Give a rigorous argument showing that $S$ is uncountable.

Ulisse Mini · Accepted Answer

We flip every bit in the diagonal just like with $\mathbf{R}$. Another way would be to show $S \sim \mathbf{R}$ by writing real numbers in base 2.

Exercise 1.6.4

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Exercise 1.6.4

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