### Real Analysis Problem: Measure Zero

Show that a countable set in R^n is of measure zero. Please see attached for full problem.

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Show that a countable set in R^n is of measure zero. Please see attached for full problem.

Please see the attached file for the fully formatted problems.

Limits of sequence. See attached file for full problem description.

Let be defined by . Use lower integral and upper integral to determine the Riemann integrability of f on [0,1]. Please see the attached file for the fully formatted problems.

If f is a function from R to R, and there exists a real number aE(0,1) such that |f'(x)|≤a for all xER , show that the equation x = f(x) has a solution.

Assume that g(t) is continuous on [a,b], K(t,s) is continuous on the rectangle a≤t, s≤b and there exists a constant M such that (a≤s≤b). Then the integral equation has a unique solution when . Please see the attached file for the fully formatted problems.

If E is equicontinuous in C(X,R), show that E-bar (the closure of E) is also equicontinuous. keywords: equicontinuity

Assume that f is a continuous real valued function on the compact space X, then show there exists a point x-bar E X such that f(x-bar)=inf{f(x): x E X). See the attached file.

Test for convergence or divergence 1.) sum from n=1 to infinity of (e^1/n)/(n^2) 2.) sum from j=1 to infinity of (-1)^j * ((sqrt j)/(j+5)) 3.) sum from n=2 to infinity of (1/((1+n)^(ln n)) keywords: tests

If f is a function from R to R which is increasing on [a,b], show that f is Riemann integrable on [a,b].

Let X be the set of all continuous functions from I_1=[t_0-a_1, t_0+a_1] into the closed ball B[g(t_0);b] is a subset of R_n. Show that for each a>0 the rule d(x,y)=max(|x(t)-y(t)|e^(-a|t-t_0|)) defines a metric on X and that the metric space (X,d) is complete.

Prove the following generalization of the Banach Fixed Point Theorem: If T is a transformation of a complete metric space X into itself such that the nth iterate, T^n, is a contraction for some positive integer n, then T has a unique fixed-point.

Newton's Method: Consider the equation f(x)=0 where f is a real-valued function of a real variable. Let x_0 be any initial approximation of the solution and let x_(n+1)=x_n - (f(x_n)/f'(x_n)). Show that if there is a positive number "a" such that for all x in [x_0-a, x_0+a] |(f(x)f''(x))/((f'(x))^2)|<=lambda<1 and |(f(x

Show that a contraction is continuous.

Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.

Let (X,d) be the metric space consisting of m-tuples of real numbers with metric d(x,y)=max{|a_k-b_k|:k=1...m} where x={a_1, a_2,...,a_m} and y={b_1, b_2,...,b_m}. In this space is every closed and bounded set compact? keywords: Heine-Borel, Borel

Show that if f is a continuous real-valued function on the compact space X, then there exist points x_1, x_2 in X such that f(x_1)=inf{f(x):x in X} and f(x_2)=sup{f(x):x in X}.

Show that a set E in the metric space X is bounded if and only if, for some "a" in X, there exists an open ball B(a;r) such that E is a subset of B(a;r).

Let X be the set of all bounded sequences of real numbers. If x=(a_k) and y=(b_k) let d be the metric funtion defined by d(x,y)=sup{|a_k - b_k|} (note _ denotes subscript) Show that the metric space defined above is complete.

Let (X,d) be a metric space. Define a closed ball with center x and radius r to be the set B[x;r]={y:d(x,y)<=r}. Prove that B[x,r] is a closed set.

Show that a convergent sequence in a metric space has a unique limit.

(See attached file for full problem description) 1. In the metric space show that: a. Any open interval of the form (a,b), (a, ), or (- ,b) is an open set. b. A close interval [a,b] is a closed set. c. Any interval of the form [a, ) is a closed set.

(See attached file for full problem description) Give an example of sets A and B in a metric space such that but d(A,B)=0.

(See attached file for full problem description) 7. If d is a real-valued function on which for all x, y, and z in X satistifes d(x,y) = 0 if and only if x=y d(x,y)+d(x,z)≥d(y,z) show that d is a metric on X.

(See attached file for full problem description) Let A be any set and let X be the set of all bounded real-valued functions defined on A. Show that defines a metric on X.

(See attached file for full problem description) Let X be the set of all bounded sequences of real numbers. If and are elements of X, show that the function d defined by is a metric on X.

Let w, x, y, z be four points in a metric space. Establish the quadrilateral inequality (see attached).

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(See attached files for full problem description) 1. Find the limit: lim(t-->0) t^2/(1-cost) 2. solve the following trigometric equation tan(2x) = 2sin(x), where 0<=0< 360 degrees

Prove that the series Sigma (k = 0 to inf) k!/k^k converges.