**//Review//** **Limit Proofs of Sequences** . A very helpful technique for proving the limit of some sequence where it is really hard to isolate n can be the following: \\ //Example: say you want to prove the limit of $\frac{4n^3 + 3n}{n^3 - 6} => \frac{3n+24}{n^3 - 6} = 4 $ . It would be hard to isolate $n$ to find such an $N > 0$. \\ Hence bound the sequence by something easier i.e. $3n + 24 \leq 27n$. Thus $\frac{27n}{n^3 - 6} < \epsilon$. Building off this technique we know the denominator $n^3 - 6 \geq \frac{n^3}{2}$ . Hence $\frac{27n}{n^3 / 2} < \epsilon => \frac{54}{n^2} < \epsilon$ .Then from here you can easily find an $N > 0$ s.t. the limit definition holds. This technique is displayed in Ross page 40 - 41.// \\ //See Ross textbook chapter 9 for useful limit properties and theorems.//\\ **Sequences**\\ **Topology** . The empty set is considered to be finite \\ . **Thm 2.6 ~ Rudin** Although a finite set cannot be equivalent to one of its proper subsets, this is possible for infinite sets. \\ //Definition of a Metric Space (Condensed - ish)//: \\ $(a)\ d(p,q)\ >\ 0\ if\ p \neq \ q; \ d(p,p) = 0$ \\ $(b)\ d(p,q)\ =\ d(q,p)$ \\ $( c )\ d(p,q)\ \le \ d(p,r)\ +\ d(r,q),\ for\ any\ r\ \in \ X$ \\ $d(x,y)\ =\ |x-y|\ (x,y\ \in \ \R^k \ )$ \\ . A K-cell is convex and compact\\ . **Thm 2.19 ~ Rudin** Every neighborbood is an open set\\ . **Thm 2.23 ~ Rudin** A set $E$ is open iff the complement is closed. Similarly with closed sets.\\ . **Thm 2.24 ~ Rudin** \\ a) The infinite union of open sets is open.\\ b) The infinite intersection of closed sets is closed. \\ c) The finite intersection of a collection of open sets is an open set. \\ d) The union of a finite set of closed sets is also a closed set. \\ //Some Statements About Compactness:// . According to Rudin, an open cover of a set $E$ in a metric space $X$ is a collection of open sets {$G_n$} such that $E\ \subset \ \cup _n G_n$. \\ . By definition, a subset $K$ of a metric space $X$ is compact if every open cover of K contains a finite subcover. \\ . Every finite set is comapact. \\ . **Thm 2.34 ~ Rudin** Compact subsets of metric spaces are closed. \\ . **Thm 2.35 ~ Rudin** Closed subsets of compact sets are compact. \\ . **Thm 2.35 corr. ~ Rudin** If $f$ is closed and $K$ is compact, then $F \cap K$ is compact.\\ . **Thm 2.37 ~ Rudin** If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$. \\ . **Thm 2.42 ~ Rudin** Every bounded, infinite set in $\R ^k$ has a limit point in $\R ^k$. \\ //Some Statements About Connected Sets:// . By Definition (Rudin), two subsets $A$ and $B$ of a metric space $X$ are said to be //separated// if both $A \cap \overline B$ and $\overline A \cap B$ are empty. \\ . A set is said to be connected if it is //not// a union of two separated sets. \\ ** Continuity** //Some Definitions of Continuity// . Suppose $X$ and $Y$ are metric spaces, $E \subset X, \ p \in \ E$, and $f$ maps $E$ into $Y$: > Then $f$ is said to be //continuous// at $p$ if for every $\epsilon \ >\ 0$ there exists a $\delta \ >\ 0$ such that $$d_y (f(x), f(p)) < \epsilon$$ for all points $x \in E$ for which $d_x (x,p) < \delta$. \\ > $f$ is said to be //uniformly continuous// on $X$ if for every epsilon greater than zero, there exists a delta such that $$d_y (f(p), f(q)) < \epsilon$$ for all $p$ and $q$ in $X$ for which $d_x(p,q) < \delta$. \\ //Important Theorems Regarding Continuity// . **Theorem 4.8 ~ Rudin** A mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous on $X$ iff $f^{-1} (V)$ is open in $X$ for every open set $V$ in $Y$ . Similarly for every closed set $C$ in $Y$.\\ . **Thm 4.14 ~ Rudin** if $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$, then $f(x)$ is compact. \\ . **Thm 4.17 ~ Rudin** if $f$ is a continuous one to one mapping of a compact space $X$ ONTO a metric space $Y$, then the inverse mapping $f^{-1}$ is a continuous mapping of $Y$ onto $X$. \\ . **Thm 4.19 ~ Rudin** If $f$ is a continuous mapping of compact metric space $X$ INTO a metric space $Y$, then $f$ is uniformly continuous on $X$. \\ . **Thm 4.22 ~ Rudin** If $f$ is a continuous mapping of a metric space $X$ INTO a metric space $Y$, an dif $E$ is a connected subset of $X$, then $f(E)$ is connected.\\ **Taylor Series** Mean Value Theorem:\\ a) If $f(a) = f(b),\ then\ \exists c\ \in (a,b)\ |\ f'( c ) = 0$ \\ b) Generalized MVT (in book) \\ c) Common MVT -> $f(a) - f(b) = (a - b)f'( c )$ \\ **//Questions//** **Taylor's Theorem and Integrability** 1) I can't quite digest the idea of what theorem 6.10 (pg. 126 Rudin) is explaining. I understand the general idea where you have some $\Delta x_i < \delta$ since you have a finite interval of discontinuous points at $f$ . Hence removing this interval of discontinuous points leaves compact, integrable sets of points along the given interval $[a,b]$. However, I don't quite understand why this idea is enough to show that a function in integrable. What if you have the function $f(x)\ =\ \mid 1/(x - 1) \mid , f(1) = 1 $ where $f$ has a single discontinuous point that is defined but the function itself is not Reimann integrable? 2) Visually speaking, I don't quite understand how a refined partition an be a union of two sub-partitions. What exactly would that look like? **Topology** 1) Aside from the midterm question, the set $\mathbb{R}$, and the empty set, what are some other examples of open and closed sets? 2) **Limits**\\ 1) How can we know/ differentiate certain approaches to particular limits of sequences that satisfy L'Hopital criteria and yield an answer but it is in fact the wrong answer?