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: $0.1234123412341234 \dots \in S$ while $0.556655665566 \dots \notin S$

Answer

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Answer

Therefore, the set $S$ is uncountable and infinite. $S is uncountable and infinite$

Steps

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: $0.1234123412341234 \dots \in S$ while $0.556655665566 \dots \notin S$

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. $S is uncountable and infinite$

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…∈S while 0.556655665566…∉S

Answer

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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: $0.1234123412341234 \dots \in S$ while $0.556655665566 \dots \notin S$