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--- math104-s21:hw3 [2021/02/06 01:02]
pzhou created
+++ math104-s21:hw3 [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:hw3 to math104-s21:hw3
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-====== HW3 ======
+====== HW 3 ======
 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/theorems that we talked about in class, and then try to prove them yourselves, then compare with notes and textbook.
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   * (b) Show that  $\limsup s_n = \inf_{N} \sup_{n \geq N} s_n$ .
+. 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.
+. 10.6
+. 10.7
+. 10.8
+. 10.11
+. 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  $t$ .

Lecture Notes