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math104-s21:hw3 [2021/02/06 01:19] pzhou |
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In this week, we finished section 10 on monotone sequence and Cauchy sequence, and also touches a bit on constructing subsequences. It is important to understand the statements of the propositions/ | In this week, we finished section 10 on monotone sequence and Cauchy sequence, and also touches a bit on constructing subsequences. It is important to understand the statements of the propositions/ | ||
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6. 10.11 | 6. 10.11 | ||
- | 7. Let $S$ be the subset of $(0,1)$ where $x \in S$ if and only if $x$ has a finite decimal expression $0.a_1 a_2 \cdots a_n$ for some $n$, and the last digit $a_n=3$. Show that for any $t \in (0,1)$ and any $\epsilon >0$, there is a sequence $s_n$ in S that converges to $t$. | + | 7. Let $S$ be the subset of $(0,1)$ where $x \in S$ if and only if $x$ has a finite decimal expression $0.a_1 a_2 \cdots a_n$ for some $n$, and the last digit $a_n=3$. Show that for any $t \in (0,1)$, there is a sequence $s_n$ in S that converges to $t$. |