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Ma221 Analysis 2

Lecturer: Dr. Derek Kitson, Dr. Rupert Levene

Website: Link

This course is a continuation of 121.

For 2007-2008, the first half of the course was lectured by Dr David Wilkins, and the second half by Dr Frederic Jaeck.

Lecture notes for a previous version of the course by Prof. David Simms are available at http://www.maths.tcd.ie/~simms/

Revision of Analysis 1

Proposition 1.1: If S is a non-empty subset of $\mathbb{R}$ which is bounded below then S has a greatest lower bound.
Proposition 1.2: $\mathbb{N}$ is not bounded above.
Proposition 1.3: Between any two real numbers there is a rational number. This is related to the idea of a dense set (later in the course).
Sequences: Formally, a sequence of reals is a map $S:\mathbb{N} \to \mathbb{R}$ . In practice, we usually write $(s_{n}):=s(n)$ . A sequence $(s_{n})$ converges to a real number if given any positive real $\epsilon > 0$ there exists a natural number $N$ such that $|s_{n}-L| < \epsilon$ for all $n>N$ . We write $\lim \limits_{n \to \infty } s_{n} = L$ and $L$ is called the limit of the sequence.
Proposition 1.4: If a sequence converges then the limit is unique.
Monotonic Increasing: A sequence $(s_{n})$ is monotonic increasing if $s_{n+1} \leq s_{n}$ for each $n$ . To check if a sequence converges we use the following:
Proposition 1.5: A bounded monotonic sequence converges.
A sequence $(s_{n})$ is called a Cauchy sequence if given $\epsilon > 0$ there exists $N$ such that

$|s_{n}-s_{m}| < \epsilon$ whenever $n,m > N$

Proposition 1.6: A sequence of real numbers converges iff it is a Cauchy sequence.
Proposition 1.7 (Bolzano-Weierstrass): Every bounded sequence of reals has a convergent subsequence.

The fundamental concepts of Calculus are limits, continuity, differentiation and integration. Recall that if $f:\mathbb{R} \to \mathbb{R}$ and $L \in \mathbb{R}$ , then we say that $f(x)$ tends to $L$ as $x$ approaches $a \in \mathbb{R}$ , if given any $\epsilon > 0$ there exists a $\delta \in \mathbb{R}$ such that:

$0 \neq |x-a| < \delta \Rightarrow |f(x)-L| < \epsilon$

We say that f is continuous at $a$ if $\lim \limits_{x \to a} f(x) = f(a)$ .

Proposition 1.8: A function is continuous iff $\lim \limits_{n \to \infty} f(x_{n}) = f(a)$ for all sequences $(x_{n})$ of real numbers converging to $a$ .
Examples of Continuous Functions:

The following result, known as the Intermediate Value Theorem, is related to the idea of connectedness.

Theorem 1.9: Let $f:[a,b] \to \mathbb{R}$ be a continuous function. Suppose $c$ is between $f(a)$ and $f(b)$ . Then there exists $x \in (a,b)$ such that $f(x) = c$ .
Applications of IVT: The Intermediate Value Theorem can be used to locate the roots of an equation. It can also be used to show that a strictly increasing continuous function $f:[a,b] \to f:[c,d]$ (with $f(a)=c$ , $f(b)=d$ ) has a continuous inverse.
Extreme Value Theorem: An optimization problem is generally concerned with finding the best way of doing something, eg. maximising are or volume or minimizing cost etc. To solve these problems we first find maxima and minima for a given interval. The Extreme Value Theorem gives sufficient conditions for a solution to such an optimization problem to exist.
Theorem 1.10 (Extreme Value Theorem): Let $f:[a,b] \to \mathbb{R}$ be a continuous function. Then $f$ has a global maximum and minimum. This relates to compactness.
Proposition 1.11: If $f:[a,b] \to \mathbb{R}$ is a continuous function then given any $\epsilon > 0$ there exists $\delta > 0$ :