Lecture Notes

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?