Differences

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

--- math104-s22:notes:lecture_2 [2022/01/19 17:54]
pzhou created
+++ math104-s22:notes:lecture_2 [2026/02/21 14:41] (current)
@@ 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