Real analysis cheat sheet, Study Guides, Projects, Research of Mathematics

Download Real analysis cheat sheet and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! Math 149s: Analysis Cheat Sheet Matthew Rognlie October 7, 2009 1 Definitions If S is some set of real numbers: 1. supS is the least upper bound of S. 2. inf S is the greatest lower bound of S. S may or may not contain its sup or inf; if it does, we say that the sup is its maximum and the inf is its minimum. For a sequence {an}∞n=1, we also define: 1. lim sup an = infk supn{an}∞n=k 2. lim inf an = supk infn{an}∞n=k We can define the limit lim an of a sequence in two equivalent ways: 1. The limit is defined if the lim inf and lim sup of the sequence exist and have the same value, in which case lim an = lim inf an = lim sup an. 2. lim an = c if for any ε > 0, we can find some N such that for all n ≥ N , |an − c| < ε. lim an = ∞ if for any y ∈ R there is N such that for all n ≥ N , an > y. We say that an infinite series ∑∞ n=1 bn converges if the limit of its partial sums limk→∞ ∑k n=1 bn converges as a sequence. There are also two equivalent notions of the limit of a function f(x) as x → y: 1. limx→y f(x) = c if for all sequences xn → y, f(xn) → c. 2. limx→y f(x) = c if for every ε > 0, we can find some δ > 0 such that for all x such that |x− y| < δ, |f(x)− c| < ε. A function f is continuous at point y if limx→y f(x) = f(y). Using our two definitions of limits, we can write this as: 1. f is continuous at y if for any sequence xn → y, f(xn) → f(y). 2. f is continuous at y if for any ε > 0, we can find some δ > 0 such that for all x such that |x− y| < δ, |f(x)− f(y)| < ε. 1 A function f that is continuous at x is differentiable at x if the limit limh→0 f(x+h)−f(x) h exists and is finite. If so, the limit is labeled f ′(x). A subset A ⊂ R is: 1. Open if for any point x ∈ A, we can find some δ > 0 such that the set B = {y : |y− x| < δ} is a subset of A. 2. Closed if for any sequence xn → x, where all xn ∈ A, x ∈ A as well. 3. Bounded if supx,y∈R |x− y| < ∞. 4. Compact if it is closed and bounded. The complement of an open set is closed, and vice versa. 2 Facts Some facts about sequences include: 1. Squeeze Theorem: (a) If an ≤ cn ≤ bn for all n, an → L and bn → L, then cn → L as well. (b) If an ≤ bn for all n and an →∞, then bn →∞ as well. 2. Cauchy Criterion: an → a if and only if for any ε > 0 we can find some N such that for all m,n ≥ N , |am − an| < ε. 3. Weierstrass Theorem: A monotonic bounded sequence converges. 4. Sequential Compactness: A compact subset of the reals is also sequentially compact, meaning that any sequence in it contains a convergent subsequence. 5. Cezaro-Stolz Theorem: Let {xn} and {yn} be two sequences of real numbers, where the yn are positive, strictly increasing, and unbounded. If limn→∞ xn+1−xn yn+1−yn = L then lim xn yn exists and is equal to L. 6. Cantor’s Nested Intervals Theorem: If I1 ⊃ I2 ⊃ . . . is a decreasing sequence of closed intervals with lengths converging to zero, then ∩∞n=1In consists of one point. Two types of series are especially important: 1. The geometric series ∑k n=0 xn has sum 1−xk 1−x . Taking k → ∞, the series converges iff |x| < 1, in which case the sum if 1 1−x . 2. The p-series ∑∞ n=0 np converges for p > 1 (assuming p is positive). You will often apply the comparison test, which states that if an, bn ≥ 0, an ≤ bn for all n and∑∞ n=0 bn converges, then ∑∞ n=0 an converges as well. If ∑∞ n=0 an diverges, then so does ∑∞ n=0 bn. The series ∑∞ n=0 an converges absolutely if ∑∞ n=0 |an| < ∞; absolute convergence implies normal convergence. Rearranging the terms of a convergent series is only guaranteed to leave the sum the same if the series converges absolutely. Some tests for absolute convergence (and convergence more generally) include: 2