Introduction to Real Analysis


1 Preliminaries

1.1 Sets and Functions


\[ R(f) \subseteq B \]

Where \(f : A \to B\).

Direct image

Say \(f : A \to B\). If \(E \subseteq A\), then the direct image of \(E\) is: \[ f(E) := \{f(x) : x \in E\} \]

Inverse image

Say \(f : A \to B\). If \(H \subseteq B\), then the inverse image of \(H\) is: \[ f^{-1}(H) := \{x \in A : f(x) \in H\} \]


One-to-one. \(x_1 \neq x_2 \implies f(x_1) \neq (x_2)\)


Complete. \(f : A \to B\) and \(f(A) = B\).


Both injunctive and complete. \[ \forall x_1, x_2 \in A. f(x_1) = f(x_2) \implies x_1 = x_2 \]

3 Sequences and Series

3.1 Sequences and Their Limits

Limit definition

A sequence \(X = (x_n)\) in \(\mathbb{R}\) converges to \(x\) if for every \(\epsilon>0\) there exists a natural number \(K(\epsilon)\) such that \(|x_{n>K(\epsilon)} - x| < \epsilon\).

Written as: \[ \lim X = x \] or \[ x_n \to x \]