### Real Analysis : Bounded Open Balls

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).

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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.

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

(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.

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

(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

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up

Find the indicated limit make sure you have a indeterminate form before you apply L'Hopital rule (1) lim xgo to0 arctan3x/arcsinx (2) lim x go to pi/2 3secx+5/tanx (3)lim x go to 0 2csc^2x/cot^2x evaluate dx/squrtpix a=0 andb=inifinity

Use only the comparison test, limit comparison test, integral test, and nth term test to test for convergence or divergence infinite series(n=2 to infinity) sqrt(25n+4/n^3-n) infinite series (n=2, infinity) ln(n)/n^3 infinite series(n=1, infinity) (n*arctan(2n))/(2n-1) *=multiply

Suppose the summation from k=1 to n of a_k is absolutely convergent and {b_n} is bounded. Prove that this implies the summation from k=1 to n of a_k*b_k is absolutely convergent.

Suppose that {a_n}_n is a real Cauchy sequence. Prove that lim superior n--> infinity (a_n) = lim inferior n--infinity (a_n) so as to conclude that lim n--infinity (a_n) exists.

Prove the following a) If lim n-->infinity (a_n*b_n) exists and lim n--> infinity (a_n) exists, then lim n -->infinity (b_n) exists. b) If lim n--> infinity (a_n) = 0 and {b_n} is bounded, then lim n-->infinity (a_n*b_n) exists and equals 0. c) If lim superior (a_n) exists, then {a_n}_n is bounded above.

Suppose a_n >0 for each n in N and lim inf (a_n) > 0. Prove there is a number a>0 st a_n >/= a for all n in N. (limit n--> infinity)

Find the domain of convergence of each of the following: a) Summation from n = 1 to infinity of [(z-1)^2n]/((2n)!) b) Summation from n = 1 to infinity of [(n^2 + 1)/(2n + 1)]z^n c) Summation from n = 1 to infinity of [(z + 1)^n]/n d) Summation from n = 0 to infinity of [(z - 1)/(z + 1)]^n

Determine if the following series converges absolutely, conditionally, or not at all. Summation from n=1 to infinity (-1)^n (n+1)/n^2

(See attached file for full problem description with equations) --- 9.3-5 Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b] such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and (1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b]. (2

(See attached file for full problem description with equations) --- 9.5-3 Without finding the sum of the series Show that --- We use the book Methods of Real Analysis by Richard Goldberg.

(See attached file for full problem description) We use the book Methods of Real Analysis by Richard Goldberg.

(See attached file for full problem description with equations) --- 94.8 Let be a sequence of functions on E such that where . Let be a nonincreasing sequence of nonnegative numbers that converges to 0. Prove that converges uniformly on E (Hint: See 3.8C) Theorem 3,8C Let be a sequence of real numbers whos

(See attached file for full problem description with equations) --- 9.4-5 Show that the series is uniformly convergent on [0,A] for any A>0. Prove that --- We are using the book of Methods of Real Analysis by Richard Goldberg.

(See attached file for full problem description with equations) --- 9.4-2 Does the series converge uniformly on (Hint: Find the sum of the series for all x) --- We are using the book of Methods of Real Analysis by Richard Goldberg.

(See attached file for full problem description with equation and proper symbols) --- 9.2-10 If be a sequence of functions that converges uniformly to the continuous function , prove that ---

6. Let M1 be a totally bounded metric space, and f: M1 --> M2 is uniformly continuous and onto. Show M2 is totally bounded. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" Please see the attached file for the fully formatted problems.

(See attached file for full problem description with proper equations) --- 3. Let T(x) = x^2 Show that T is a contraction on (0, 1/3] , but that T has no fixed point on this interval. Does this conflict Theorem 6.4? Explain. Note: We are using the book Methods of Real Analysis by Richard R. Goldberg. This

2. Prove that if a metric space M is totally bounded, then there is a countable dense subset of M. Note: we are using the "Methods of Real Analysis by Richard R Goldberg

1. Give an example of a set E such that both E and its complement are dense in R^1. Then show that such a set E can not be closed. Note: we are using the "Methods of Real Analysis by Richard R Goldberg" ---

Find the radius of convergence of the series sum from n = 1 to infinity of n^3(z/3)^n. Does this series converge at any point on the boundary of the disk of convergence?

G(x,y)=｜x｜y (that's module sign) prove that g is not differentiable at (0,b) for any value non-zero value of b.