Q4 (10 points)
Let $S$ be the subset of $(0,1)$ consisting of numbers whose digits (in the decimal expansion) alternate between odd and even (including the leading 0 ). Prove that $S$ is uncountable and infinite.
For instance: $0.1234123412341234 \ldots \in S$ while $0.556655665566 \ldots \notin S$
Answer
Therefore, the set $S$ is uncountable and infinite. \(\boxed{S \text{ is uncountable and infinite}}\)
Step 1 ：Let $S$ be the subset of $(0,1)$ consisting of numbers whose digits (in the decimal expansion) alternate between odd and even (including the leading 0 ).
Step 2 ：For instance: $0.1234123412341234 \ldots \in S$ while $0.556655665566 \ldots \notin S$
Step 3 ：We need to prove that $S$ is uncountable and infinite.
Step 4 ：This can be proven by Cantor's diagonal argument.
Step 5 ：Therefore, the set $S$ is uncountable and infinite. \(\boxed{S \text{ is uncountable and infinite}}\)
