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--- math104-s22:notes:lecture_4 [2022/01/27 09:22]
pzhou
+++ math104-s22:notes:lecture_4 [2022/01/27 10:46] (current)
pzhou [Lecture 4]
@@ Line 6: / Line 6: @@
   * Cauchy sequence.
-Discussion time: Ex 10.1, 10.6 in Ross
+Discussion time: Ex 10.1, 10.7, 10.8 in Ross
 ==== limit goes to  $+\infty$ ? ====
@@ Line 34: / Line 34: @@
 ====  $\liminf$  and  $\limsup$  ====
-Recall the definition of  $\sup$ .
+Recall the definition of  $\sup$ . For  $S$  a subset of  $\R$ , bounded above, we define  $\sup(S)$  to be the real number  $a$ , such that  $a$  is  $\geq$  than any element in  $S$ , and for any $\epsilon>0$, there is some  $s \in S$  such that  $s > a-\epsilon$ .
+Also, for a sequence  $(a_n)_{n=m}^\infty$ , we can define the 'value set'  $\{a_n\}_{n=m}^\infty$ , which is the 'foot print' of the 'journey'.
+Also, for a sequence  $(a_n)_{n=1}^\infty$ , we can define the tail  $(a_n)_{n=N}^\infty$ , and we only care about the tail of a sequence.
+We want to define a gadget, that captures the 'upper envelope' of a sequence, what does that mean? Let  $(a_n)$  be a seq, we want define first an auxillary sequence
+ $A_m = \sup_{n \geq m} a_n$
+then we define
+ $\limsup a_n = \lim A_m (= \inf A_m)$
+Time for some examples,  $a_n = (-1)^n (1/n)$ .

Lecture Notes