Differences

This shows you the differences between two versions of the page.

--- math104-s22:notes:lecture_2 [2022/01/19 09:54]
pzhou created
+++ math104-s22:notes:lecture_2 [2022/01/19 10:13] (current)
pzhou [Group Discussion (20min)]
@@ Line 11: / Line 11: @@
 Next, we briefly mention what the symbol  $+\infty$  and  $-\infty$  mean. Note that, these are not real numbers, they are not member of  $\R$ . We introduce them to simplify certain statement of results. For example, we can now say, given any subset  $E \In \R$ , the  $\sup(E)$  exists in  $\R \cup \{+\infty\}$ . (What's wrong with this expression  $\R \cup +\infty$ ? Why the curly braces? )
-That hopefully will take us 40 min. We will use the last 30 min to talk about sequence and limits in  $\R$ , this is Ross Sec 7 and Tao-I Ch 6. So, what does limit mean? We say a sequence (of real numbers)  $(a_n)$  converges to  $a$ , if for any $\epsilon>0$, there exists  $N>0$ , such that for any  $n > N$ , we have  $|a_n - a| < \epsilon$ . Informally, we say, for any  $\epsilon$ , the sequence eventually fell into the  $\epsilon$ -neighborhood of  $a$ .
+That hopefully will take us 40 min. We will use the last 20 min to talk about sequence and limits in  $\R$ , this is Ross Sec 7 and Tao-I Ch 6. So, what does limit mean? We say a sequence (of real numbers)  $(a_n)$  converges to  $a$ , if for any $\epsilon>0$, there exists  $N>0$ , such that for any  $n > N$ , we have  $|a_n - a| < \epsilon$ . Informally, we say, for any  $\epsilon$ , the sequence eventually fell into the  $\epsilon$ -neighborhood of  $a$ .
+Let's finish by go through some examples of convergence, just to test how the definition works.
+[[https://courses.wikinana.org/_media/math104-s21/note_jan_21_2021_3_.pdf | The note from previous semester]] might be useful.
+===== Group Discussion (20min) =====
+. Let  $A, B \In \R$ , and define  $A+B = \{a + b| a \in A, b \in B \}$ . Prove that  $\sup (A+B) = \sup A + \sup B$ .
+. Same setup as above, prove that  $\sup (A \cup B) = \max(\sup A, \sup B)$ .
+. Ross 7.1, 7.2, 7.3
+{{:math104-s22:notes:pasted:20220119-101341.png}}

Lecture Notes

User Tools

Site Tools

Differences

Page Tools