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--- math104-s22:notes:lecture_4 [2022/01/27 17:22]
pzhou
+++ math104-s22:notes:lecture_4 [2026/02/21 14:41] (current)
@@ 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