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Real Analysis

Proof Regarding Continuous Functions

Let f be a function defined on all of R that satisfies the additive condition f(x+y)=f(x)+f(y) for all x,y belong to R a- Show that f(0)=0 and that f(-x)=-f(x) for all x belong to R. b- Show that if f is continuous at x=0 then f is continuous at every point in R c- Let k=f(1) show that f f(n)=kn for all n belong to N and

Proof Regarding Continuity and Contraction Mapping

(contraction mapping theorem).let f be a function defined on all of R and assume there is a constant c such that 0<c<1 and Absolute value of f(x)-f(y)<= c Absolute value of x-y for all x,y belong to R show that f is continuous on R.

Real Analysis

A- Show that if a function is continuous on all of R and equal to 0 at every rational point then it must be identically 0 on all of R b- if f and g are continuous on all of R and f(r)=g(r) at every rational point,must f and g be the same function?

Real Analysis

Assume h:R->R is continuous on R and let K={x:h(x)=0}. show that K is a closed set.

Real Analysis

Let g:A->R and assume that f is a bounded function on A subset or equal to R (i.e there exist M>0 satisfying Absolute value of f(x)<=M for all x belong to A). Show that if lim g(x)=0 as x->c, then g(x)f(x)=0 as x->c as well.

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

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

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

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.

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

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.

Real Analysis

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

Real Analysis: Cauchy Sequences

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.

Real Analysis : Converging Sequences

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 that (a_n) must converges to a.

Real Analysis: Sequences

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

Real Analysis: Countable Sets and Antichains

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

Real Analysis : Empty Set and Nested Interval Property

Prove that: Intersection to infinity for n=1 (sign of intersection with infinity on top and n=1 in the bottom) of (0,1/n)=empty. (Notice that this demonstrates that the interval in the Nested Interval Property must be closed for the conclusion of the theorm to hold.)