math104-s21:hw3
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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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| * (b) Show that $\limsup s_n = \inf_{N} \sup_{n \geq N} s_n$. | * (b) Show that $\limsup s_n = \inf_{N} \sup_{n \geq N} s_n$. | ||
| + | 2. Let $(a_n), (b_n)$ be two bounded sequences, show that | ||
| + | $$\limsup (a_n + b_n) \leq \limsup(a_n) + \limsup(b_n)$$ | ||
| + | and give an example where the inequality is strict. | ||
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| + | 3. 10.6 | ||
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| + | 4. 10.7 | ||
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| + | 5. 10.8 | ||
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| + | 6. 10.11 | ||
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| + | 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 $t$. | ||
math104-s21/hw3.1612602142.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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