MTH 303: Real Analysis Semester 1, 2013-14 1. Real numbers 1.1. Decimal and ternary expansion of real numbers. (i) [0; 1] and R are uncountable. (ii) Cantor set is an uncounbtable set of measure zero. 1.2. Order properties of real numbers. (i) Least upper bound axiom. (ii) Every subset of R that has a lower bound also has a greatest lower bound. (iii) Archimedean property. (iv) Density of Q. 2. Sequences of real numbers 2.1. Limits of sequences. (i) Sequences and their limits with examples. (ii) A sequence in R has at most one limit. (iii) A sequence converges if and only if its tail converges. (iv) Let (xn) be a sequence of real numbers and let x 2 R. If (an) is a sequence of positive reals with lim(an) = 0 and if we have a constant C > 0 and some m 2 N such that jxn − xj ≤ Can, for n ≥ m, then lim(xn) = x. 2.2. Limit Therorems. (i) Bounded sequences with examples. (ii) A convergent sequence of real numbers is bounded. (iii) Convergence of the sum, difference, product, and quotient of sequences. (iv) The limit of a non-negative sequence is non-negative. (v) Squeeze Theorem: Suppose that X = (xn), Y = (yn), and Z = (zn) are sequences such that xn ≤ yn ≤ zn, 8 n, and lim(xn) = lim(zn). Then Y converges and lim(xn) = lim(yn) = lim(zn). (vi) Let X = (x ) be a sequence of positive reals such that L = lim xn+1 . n xn If L < 1, then X converges and lim(xn) = 0. 1 2 2.3. Monotone Sequences. (i) Monotone sequences with examples. (ii) Monotone Covergence Theorem: A sequence of real numbers is conver- gent if and only if it is bounded. Moreover, (a) If X = (xn) is a bounded increasing sequence, then lim(xn) = supfxn j n 2 Ng. (b) If Y = (yn) is a bounded decreasing sequence, then lim(yn) = inffyn j n 2 Ng. 2.4. Subsequences. (i) Subsequences with examples. 0 (ii) If a sequence X = (xn) converges to x, then any subsequence X = (xnk) of X also converges to x. (iii) Divergence Criteria: If a sequence X = (xn) of real numbershas either of the two properties, then X is divergent. (a) X has 2 convergent subsequences converging to different limits. (b) X is unbounded. (iv) Monotone Subsequence Theorem: If X = (xn) is a sequence of real numbers , them there exists a subsequence of X that is monotone. (v) Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence. (vi) If X = (xn) is a bounded sequence of reals such that every subsequence of X converges to x 2 R, the X converges to x. 2.5. The Cauchy Criterion. (i) Cauchy sequences with examples. (ii) Every convergent sequence is Cauchy. (iii) Every Cauchy sequence is bounded. (iv) Completeness property: Every Cauchy sequence is convergent. (v) Contractive sequences. (vi) Every contractive sequence is Cauchy. 2.6. Properly divergent sequences. (i) Properly divergent sequences with examples. (ii) A monotone sequence of real numbers is properly divergent if and only if it is unbounded (iii) Let (xn) and (yn) be sequences such that xn ≤ yn, for all n. (a) If lim(xn) = +1, then lim(yn) = +1. 3 (b) If lim(yn) = −∞, then lim(xn) = +1. (iv) Let (xn) and (yn) be sequences of positive real numbers such that lim(xn=yn) = L, for some L > 0. Then lim(xn) = +1 if and only if lim(yn) = +1. 2.7. Limit inferior and superior. (i) Subsequence limit (ii) Limit inferior and superior (iii) A bounded sequence (xn) is convergent if and only if lim sup(xn) = lim inf(xn). 3. Sequences of functions 3.1. Pointwise and uniform convergence. (i) Pointwise convergent sequences with examples. (ii) Uniformly convergent sequence with examples. (iii) Criterion for non-uniform convergence. (iv) The uniform norm k kA. (v) A sequence (fn) of bounded functions on A ⊆ R converges uniformly on A to f if and only if kfn − fkA ! 0. (vi) Cauchy criterion for uniform convergence: A sequence of bounded functions on A ⊆ R converges uniformly on A to a bounded function f if and only for each > 0 there exists a number K 2 N such that for all m; n ≥ K, then kfm − fnkA ≤ . (vii) Uniform Limit Theorem: Let (fn) be a sequence of continuous function on a set A ⊆ R such that fn converges to f uniformly on A to a function f. Then f is continuous on A. (viii) Interchange of limit and derivative: Let (fn) be a sequence of functions on a bounded interval J ⊂ R. Suppose that fn converges pointwise at 0 x0 2 J and that (fn) exists on J and converges uniformly to a function 0 g. Then (fn) converges uniformly on J to a function f such that f = g. (ix) Compactness in R. (x) Dini's Theorem: if (fn) is a sequence of continuous functions on [a; b] that converges on [a; b] to a continuous function f. Then the conver- gence of the sequence is uniform. 3.2. The metric space C[a; b]. (i) Brief introduction to metric spaces and norms. (ii) The supremum norm and the metric it induces in C[a; b]. 4 (iii) Convergence with respect to the supremum norm is equivalent to uniform convergence. (iv)C[ a; b] is complete. (v) Boundedness and total boundedness in metric spaces. (vi) A totally bounded subset of a metric space is bounded. (vii) -dense subsets of a metric space. (viii) A subset A of metric space (X; d) is totally bounded if and only if for every > 0, there exists a finite subset of A that is -dense in A. (ix) A subset A of a metric space (X; d) is totally bounded if and only if every sequence in A has a Cauchy subsequence. (x) Equicontinuous families of functions. (xi) Ascoli-Arzela Theorem: Let F be a bounded equicontinuous subset of the metric space C[a; b]. Then F is totally bounded. (xii) Every bounded equicontinuous sequence of functions in C[a; b] has a uniformly convergent subsequence. (xiii) Weierstrass Approximation Theorem: Let f be any function in C[a; b]. Then given > 0, there exists a polynomial P such that kP − fk < . In other words, the set P of polynomials is dense in C[a; b]. 4. Series of functions 4.1. Convergence and uniform convergence of a series of functions. (i) Convergence and uniform convergence with examples. (ii) Uniform Limit Theorem: Let (fn) be a sequence of real-valed function P1 on a metric space X. If n=1 fn converges uniformly to f on X and if each fn is continous at a 2 X, then f is also continuous at X. P1 (iii) Weierstrass M-test: Let n=1 fn be a series of real-valued functions on a set E. If there exists a sequence (Mn) of positive real numbers P1 P1 P1 with k=1 Mk < 1 such that k=1 fk(x) k=1 Mk, for all x 2 E, P1 then k=1 fk converges uniformly on E. P1 k (iv) If the power series k=0 akx coverges for x = x0, then it converges uniformly on [−a; a], where a is any number such that 0 < a < jx0j. P1 (v) Dini's Theorem: Let n=1 fn be a series of continuous nonnegative P1 functions on the a compact metric space X. If n=1 fn converges on X to the continuous function f, then the convergence is uniform on X. 4.2. Taylor's Theorem. (i) Taylor series with examples. 5 (ii) Taylor's Theorem (with intergal form of remainder): Let f be a real valued function on [a; a + h] such that f (n+1)(x) exists for every x 2 [a; a + h] and f (n+1) is continuous in [a; a + h]. Then f 0(a) f 00(a) f(x) = f(a) + (x − a) + (x − a)2 + ::: 1! 2! f (n)(a) + (x − a)n + R (x 2 [a; a + h]); n! n+1 where Z x 1 n (n+1) Rn+1 = (x − t) f (t) dt : n! a (iii) Taylor's Theorem (with Lagrange form of remainder): Let f be a real valued function on [a; a + h] such that f (n+1)(x) exists for every x 2 [a; a+h] and f (n+1) is continuous in [a; a+h]. Then if x 2 [a; a+h], there exists a number c 2 [a; x] such that f 0(a) f 00(a) f (n)(a) f(x) = f(a) + (x − a) + (x − a)2 + ::: + (x − a)n 1! 2! n! f (n+1)(c) + (x − a)n+1: (n + 1)! (iv) Taylor's Theorem (with Cauchy form of remainder): Let f be a real valued function on [a; a + h] such that f (n+1)(x) exists for every x 2 [a; a + h] and f (n+1) is continuous in [a; a + h]. Then if x 2 [a; a + h], there exists a number c 2 [a; x] such that f 0(a) f 00(a) f (n)(a) f(x) = f(a) + (x − a) + (x − a)2 + ::: + (x − a)n 1! 2! n! f (n+1)(c) + (x − c)n(x − a): n! (v) The series expansions for the elementary functions log(1 + h) and ex. 5. Functions of bounded variation 5.1. Properties of monotonic functions. (i) Let f be an increasing function defined on [a; b] and let and let P = fx0; x1; : : : ; xng be a partition of [a; b].