### 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.

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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.

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.

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.

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.

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.

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

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.

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

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?

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.

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

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

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

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.

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

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

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.

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

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?

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

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))

Show that if the series sum(sum sign) to infinity(top) of k=1(bottom) of a_k converges then a_k--->0

Show that if sum(sum sign) to infinity(top) of k=1(bottom) of a_k=A and sum(sum sign) to infinity(top) of k=1(bottom) of b_k=B, then 1-sum(sum sign) to infinity(top) of k=1(bottom) of ca_k=cA for all c belong to R 2-sum(sum sign) to infinity(top) of k=1(bottom) of (a_k+b_k)=A+B

Assume a_n and b_n are Cauchy sequences.Use a triangle inequality argument to prove c_n=Absolute value of a_n-b_n is Cauchy.

Show that the series sun sign over it infinity sign and below it n=1 of 1/n^p converges if and only ifp>1

Let (a_n) be a bounded sequence and define the set S={x belong to R: x< a_n for infinitely many terms a_n}. show that there exists a subsequence(a_nk) converging to s=sup S

Give an example of each of the following or argue that such a request is impossible: 1) A Cauchy sequence that is not monotone. 2) A monotone sequence that is not Cauchy. 3) A Cauchy sequence with a divergent subsequence. 4) An unbounded sequence containing a subsequence that is Cauchy.

Assume (a_n) is a bounded sequence with the property that every convergent subsequence of (a_n) converges to the same limit a belong to R.show that (a_n) must converges to a.

Give an example of each of the following, or argue that such a request is impossible: 1) A sequence that does not contain 0,1 as a term but contains subsequences converging to each of these values. 2) A monotone sequence that diverges but has a convergent subsequence. 3) A sequence that contains subsequences converging to

Answer the following by establishing 1-1(one to one) correspondence with a set of known cardinality: 1 - Is the set of all functions from{0,1} to N countable or noncountable? 2 - Is the set of all functions from N to {0,1} countable or noncountable? 3 - Given a set B ,a subset A of P(B) is called an antichain if no element of