### Sample Problems: 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.

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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 equation and proper symbols) --- 9.2-10 If be a sequence of functions that converges uniformly to the continuous function , prove that ---

Let {fn} infinity-->n-1 be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each nЄ I let Fn(x) = ∫ x--> a fn(t)dt a<x<b Show that {fn} infinity-->n-1 converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F;

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.

Prove (using the definitions of the limit at infinity and at a real number) that if lim x-> + or - ∞ (infinity) of f(x) = + or - ∞, then lim x-> + or - ∞ of 1/f(x) = 0.

As x approaches -2 from the left, what is the limit of (square root of x^2 +5) / (x+2) Please show work if you can. Choices are A. 3/2, B. 0, C. -infinity D. -1, E. + infinity

For each type of measure, give two examples of populations where it would be the most appropriate indication of central tendency. Each type of measure is for mean, and the median.

Independence and relations Real Analysis Jacobians (XXVII) Jacobian of Functions of Functions Compute the Jacobian del(u,v)/del(r, the

Independence and relations Real Analysis Jacobians (XXVI) Jacobian of Implicit Functions If lemda, mu, nu are the roots of the equa

lim [(cos x-1)x]/sin x x->0 Please see the attached file for the fully formatted problems.

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship Fn=Fn-1+Fn-2 where F1=1, F2=1. (b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1. (c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fi

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e. a) For a fixed integer n>0, let f(x) = (n+1)xn - nxn+1 . For x >1, show f is decreasing and that f(x) . Hence, for x >1; Xn(n+1-nx) < 1 b) Substitute the following x-value into the inequality from part (a)

Consider the series ∞ Σ ln (((k-1)(k+1))/k^2) = ln ((1 *3)/(2*2) + ln ((2*4)/(3*3)) +... k=2 a. Show the partial sum S4 = ln (5/8) b. Show the partial sum Sn = ((n+1)/(2n)) c. Use part b to show the partial sums Sn and therefore the series, converges.

Please see the attached file for the fully formatted problems.

Independence and relations Real Analysis Jacobians (XI) If u = (y^2)/(2x) and v = (x^2 + y^2)/(2x), find the Jacobian of u,v with respect to x,y. The fully formatted problem is in the attached file.

Find the limit and justify your answer: lim n--> ∞ ∫ 0 --> ∞ sin nt/ (1 + t^2) dt Please see the attached file for the fully formatted problems.

Suppose that {an} and {bn} are sequences of positive terms, and that the limit as n goes to infinity of (an/bn) = L > 0. Prove that limit as n goes to infinity of an is positive infinity if and only if the limit as n goes to infinitiy of bn is positive infinity. Here is what I have for proving the first way: Suppose that

1) Let M be an elementary set. Prove that | closure(M)M | = 0. (closure of M can also be written as M bar, and it is the union of M and limit points of M). 2) If M and N are elementary sets then show that | M union N | + | M intersection N| = |M| + |N| The definition of elementary set : If M is a union of finite members

If |fn(x)| < gn(x) for all nE R and every x E[a,b] , and the series... converges uniformly in [a,b], then ... converges uniformly in [a,b]. Please see the attached file for the fully formatted problems.

Independence and relations Real Analysis Jacobians (VIII) Let u = (x + y)/(1 - xy) and v = tan inverse x + tan inverse y. If xy is not equal to 1,

Prove that the sequence of functions ... converges for every , and find the limit to which it converges. Please see the attached file for the fully formatted problems.

Find Limits and Rate of Convergence.

Independence and relations Real Analysis Jacobians (IV) If u = x + y + z, v = x^2 + y^2 + z^2, w = x^3 + y^3 + z^3 - 3xyz Show that u,v,w are connected by