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# Real Analysis

### Real Analysis : Convergent and Cauchy Sequences

See attached file for all symbols. --- ? For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example. 1) every bounded sequence of real numbers is convergent. 2) Every convergent sequence is monotone. 3) Every monotone and bounded sequence of real number

### Real Analysis: Proof Regarding Continuity and Belonging to Sets

Let C be the Cantor set defined C=intersection sign on top inf bottom n=0 C_n.Define g:[0,1]->R by g(x)={1 if x belong to C and 0 if x does not belong to C. a-show that g fails to be continuous at any point c belong to C. b-prove that g is continuous at every point c does not belong to C.

### Constructing Functions with Discontinuities

For each of the following choices of A,construct a function f:R->R that has discontinuities at every point x in A and is continuous on A^c(compliment) a-A=Z b-A={x:0<x<1} c-A={x:0<=x<=1} d-A={1/n:n belong to N}.

### Real Analysis

(Composition of continuous Functions).Given f :A->R and g:B->R, assume that the range of f(A)={f(x):x belong to A} is contained in the domain of B so that the composition g o f(x)=g(f(x)) is well-defined on A.If f is continuous at c belong to A, and if g is continuous at f(c) belong to B, then g o f is continuous at c. -Supply

### Determining If Series are Convergent or Divergent

(a) A certain infinite series (some of whose terms are positive and some of whose terms are negative) is known to converge, but does not converge absolutely. Explain how this is possible, by giving two such example series. (b) Determine whether each of the attached infinite series converges or diverges. See the attached file

### Sum of a Series in the Closed Form

Please see the attached file for the fully formatted problem. Find the sum in the closed form: 1/2! + 2/3! +...+ n/(n+1)!

### Extrema and Rolle's Theorem

See attachment. 1) Locate and classify the extrema of the following functions: Justify your answer 2) Let be continuous functions, such that f(0)+g(0)=0 and f(1)+g(1)=0. assume also that f, g are also differentiable for every use Rolle's theorem to show that there exists an , such that (see the attachment).

### Euler's Identity

Recall the Taylor series Sum(x^n/n!). The same series can be used to define e^z for a complex number z=a+bi. Use the Taylor series to show that exp(iy) = cos(y) + i sin(y) for any real number y. To do this substitute iy into the series and compute several terms. Look for patterns. See the attached file.

### Real Analysis : Connected and Disconnected Sets and Closure

A- Find an example of a disconnected set whose closure is connected. b- If A is connected ,is A closure necessarily connected? If A is perfect is A closure necessarily perfect?

### Real Analysis : Limit and Sequence

Show that if x=lim a_n for some sequence (a_n) contained in A satisfying a_n not = x,then x is a limit point of A.

### Real Analysis : Open Intervals

Show that it is impossible to write R=U(union sign n=1 bottom, infinity top)F_n where for each n belong to N, F_n is closed set containing no nonempty open intervals.

### Real Analysis : Connectedness and Convergent Sequence

Show that A set E subset or equal to R is connected if and only if, for all nonempty disjoint sets A and B satisfying E=A U B there always exists a convergent sequence (x_n)-->x with (x_n) contained in one of A or B and x an element of the other.

### Real Analysis : Convergent and Divergent Summations

Give an example to show that its possible for both Sum of x_n(sum sign) and sum of y_n to diverge but for Sum of x_n y_n to converge.

### Real Analysis : Sequences and Limits

Show that if f be a function defined on A, and c be a limit point of A. If there exist two sequences (x_n) and (y_n) in A with x_n not =c y_n not = c and lim x_n=limy_n=c but lim f(x_n) not = f(y_n), then we conclude that the functional limit lim_x-->c f(x) does not exist.

### Real Analysis : Subsets and Limits

Let f and g be functions defined on a domain A subset or equal to R, and assume lim_x-->c f(x)=L and lim_x-->c g(x)=M for some limit point c of A then, 1-lim_x-->c k f(x)=kL for all k belong to R. 2-lim_x--> [f(x)+g(x)]=L+M 3-lim_x-->c [f(x)g(x)]=LM 4-lim_x-->c f(x)/g(x)=L/M provided M not = 0

### Real Analysis : Baire Category Theorem

If {G1,G2,G3,...} is a countable collection of dense, open sets then the intersection (U top infinity bottom n=1)G_n is not empty.

### Real analysis

Definition: A set A subset or equal to R is called an F_&(F sigma) set if it can be written as the countable union of closed sets. A set B subset or equal to R is called G_&(G sigma) if it can be written asthe countable intersection of open sets. 1-Argue that a set A is a G_& (G sigma) set if and only if its complement is

### Real Analysis

A set E is totally disconnected if, given any two points x,y belong to E there exist separated sets A and B with x belong to A and y belong to B and E=A U B. 1-show that Q is totally disconnected. 2-is the set of irrational numbers totally disconnected?

### Real Analysis

Let A and B be subsets of R show that if there exists disjoint open sets U and V with A subset or equal of U and B subset or equal of V then A and B are separated.

### Real Analysis

Show that if K is compact and F is closed then K intersection F is compact.

### Real Analysis

Show that if K is compact, then sup K and inf K both exist and are elements of K

### Real analysis

Let A be bounded above so that s= sup A exists show that s belong to closure A(A over it bar)

### Real analysis

Let x belong to O, where O is an open set.If (x_n) is a sequence converging to x prove that all but a finite number of the terms of (x_n) must be contained in O.

### Limit Approach Functions

Find the following: the limit as x approaches 0 of (tan 3x * cot 2x)

### Convergent Sequence of Rationals

Show that, for every real number y, there is a sequence of rational numbers which converges to y.

### Real Analysis

A set F subset or equal to R is closed if and only if every Cauchy sequence contained in F has a limit that is also an element of F.

### Real Analysis

Show that if sum x_n converges absolutely and the sequence(y_n) is bounded then the sum x_n y_n converges.

### Real Analysis

1-Show that if sum a_n converges absolutely then sum a^2_n also converges absolutely.Does this proposition hold without absolute converge. 2-if sum a_n converges and a_n>=0 can we conclude anything about sum of sqrt a_n?

### Real Analysis Divergence

Let Sum of an(sign of sum) be given.For each n belong to N let p_n=an if a_n is positive and assign p_n=0 if a_n is negative.In a similar manner,let q_n=a_n if an is negative and q_n=0 if a_n is positive. 1-Argue that if Sum a_n diverges then at least one of sum p_n or sum q_n diverges. 2- show that if sum a_n converges co

### Power series multiplication

Please show proper notation, justification and step by step work. n See attachment for problem Given that the zeros for (sinx)/x are the values x=0, x= +-pie, x=+-2pie, x=+-3pie,.... (x=0 must be excluded, why?) This implies that F(x) can be factored as follows F(x) = (1-(x/pi)) (1-(x/-pi)) (1-(x/2pi)) (1-(x/3pi))