Final Exam Cheat Sheet

1. Sequences and Limits

Convergence: A sequence \((s_n)\) converges to \(s\) if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall n > N, |s_n - s| < \epsilon\).
Supremum (\(\sup S\)): The least upper bound of set \(S\).
Infimum (\(\inf S\)): The greatest lower bound of set \(S\).
Limsup: \(\limsup s_n = \lim_{N \to \infty} (\sup \{s_n : n > N\})\).
Liminf: \(\liminf s_n = \lim_{N \to \infty} (\inf \{s_n : n > N\})\).

Squeeze Lemma: If \(a_n \le s_n \le b_n\) and \(\lim a_n = \lim b_n = s\), then \(\lim s_n = s\).
Monotone Convergence: Every bounded monotone sequence converges.
Bolzano-Weierstrass: Every bounded sequence has a convergent subsequence.
Subsequential Limits: The set of subsequential limits of a bounded sequence is closed and bounded. \(\limsup s_n\) is the largest subsequential limit.

\(\lim \frac{1}{n^p} = 0\) for \(p > 0\).
\(\lim c^{1/n} = 1\) for \(c > 0\).
\(\lim n^{1/n} = 1\).
Recursive: If \(s_{n+1} = \sqrt{s_n + 1}\), assume limit \(L\) exists, solve \(L = \sqrt{L+1}\).

Ratio Test: Let \(L = \lim |\frac{a_{n+1}}{a_n}|\).
- If \(L < 1\), converges absolutely.
- If \(L > 1\), diverges.
- If \(L = 1\), inconclusive.
Comparison Test: If \(0 \le a_n \le b_n\):
- \(\sum b_n\) converges \(\implies \sum a_n\) converges.
- \(\sum a_n\) diverges \(\implies \sum b_n\) diverges.
Alternating Series Test: \(\sum (-1)^n a_n\) converges if \(a_n\) is decreasing and \(\lim a_n = 0\).

Geometric: \(\sum r^n\) converges iff \(|r| < 1\). Sum = \(\frac{\text{first}}{1-r}\).
p-series: \(\sum \frac{1}{n^p}\) converges iff \(p > 1\).
Harmonic: \(\sum \frac{1}{n}\) diverges.

Continuity at \(x_0\): \(\forall \epsilon > 0, \exists \delta > 0\) such that \(|x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon\).
Sequential Continuity: \(f\) is continuous at \(x_0\) iff for every sequence \(x_n \to x_0\), \(f(x_n) \to f(x_0)\).

Intermediate Value Theorem (IVT): If \(f\) is continuous on \([a, b]\) and \(y\) is between \(f(a)\) and \(f(b)\), \(\exists c \in (a, b)\) such that \(f(c) = y\).
Application: Roots of polynomials, fixed points (\(f(x)=x\)).
Extreme Value Theorem: A continuous function on a closed bounded interval \([a, b]\) attains its maximum and minimum.
Algebra of Continuity: Sums, products, quotients (denom \(\ne 0\)), and compositions of continuous functions are continuous.

For \(f(x) = x^2\) at \(x_0\): \(|x^2 - x_0^2| = |x-x_0||x+x_0|\). Bound \(|x+x_0|\) by restricting \(\delta \le 1\).
For \(f(x) = \sqrt{x}\) at \(x_0 > 0\): \(|\sqrt{x} - \sqrt{x_0}| = \frac{|x-x_0|}{\sqrt{x}+\sqrt{x_0}} \le \frac{|x-x_0|}{\sqrt{x_0}}\).

Uniform Continuity: \(\forall \epsilon > 0, \exists \delta > 0\) such that \(\forall x, y \in S, |x - y| < \delta \implies |f(x) - f(y)| < \epsilon\).
Key: \(\delta\) depends only on \(\epsilon\), not on \(x\).

Heine-Cantor: A continuous function on a closed bounded interval \([a, b]\) is uniformly continuous.
Non-Uniformity: If \(f\) is unbounded on a bounded set, it is NOT uniformly continuous (e.g., \(1/x\) on \((0, 1)\)).
Lipschitz: If \(|f(x) - f(y)| \le K|x - y|\) for all \(x, y\), then \(f\) is uniformly continuous (take \(\delta = \epsilon/K\)).
Functions with bounded derivatives are Lipschitz.

\(f(x) = 3x + 11\): Uniformly continuous on \(\mathbb{R}\) (\(\delta = \epsilon/3\)).
\(f(x) = x^2\): Uniformly continuous on bounded \([0, M]\) (\(\delta = \epsilon/(2M)\)), but NOT on \(\mathbb{R}\).
\(f(x) = \sin(1/x)\): Bounded but not uniformly continuous on \((0, 1)\) (oscillates too fast).