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Real Analysis

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 : 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 R.show that (a_n) must converges to a.

Real analysis

Prove: subsequences of a convergent sequences converge to the same limit as the original sequence

Real Analysis

Show that lim inf a_n = lim sup a_n if and only if lim a_n exists

Real Analysis

Show that if x_n <= y_n <= z_n for all n belong to N and if lim x_n=limz_n=L then lim y_n=L as well

Real Analysis

A) Show that if (b_n)-->b,then the sequence of absolute values Absolute value of b_n converges to absolute value of b

Real Analysis

Show that limits, if exist, must be unique. In other words, assume lim an=L1 and lim an=L2 and prove that L1=L2

Real Analysis

Let xn(smaller n)>=0 for all n belong to N a) if (xn)-->0, show that(sqrt[xn])-->0 b)if (xn)-->x,show that(sqrt[xn])-->x

Real Analysis

Using the definition of convergence of a sequence show that the following sequences converge to the proposed limit: 1-lim 1/(6n^2+1)=0 2-lim 2/sqrt[n+3]=0 3-lim (3n+1)/(2n+5)=3/2

Real Analysis

1- if A1,A2,A3,...,Am are each countable sets, then the union A1 U A2 U A3...U Am is countable 2- if An is a countable set for each n belong to N,then Un=1(to infinity) An is countable

Real Analysis

Proof: if A subset or equal of B and B is countable, then A is either countable, finite or empty.

Real analysis

Prove that if a is an upper bound for A and if a is also an element of A, then it must be that a=sup A

Real Analysis

Assume that A And B are nonempty, bounded above and satisfy B subset or equal of A. Show that sup B<= sup A

Real Analysis

Let y1=1,and for each n belong to N define y_n+1=(3y_n+4)/4. a-Use induction to prove that the sequence satisfies y_n<4for all n belong to N. b-use another induction argument to show the sequence(y1,y2,y3,...)is increasing.

Real Analysis

Given a function f and a subset A of its domain, let f(A) represent the range of f over the set A; f(A)={f(x) : x belong to A}. a-Let f(x)=x^2. If A=[0,2](the closed interval{x belong to R : 0<=x<=2}) and B=[1,4],find f(A) and f(B).Does f(A intersection B)=f(A) intersection f(B) in this case?.Does f(A U B)=f(A) U f(B)?. b-Fi

Real Analysis

Please see the attached file for the fully formatted problems. Suppose that is not a perfect nth power, i.e K is not equal to (a) Prove that is not a member of Q, the set of all rational numbers. (b) Infer that the nth root of a natural number is either a natural number or it is irrational.

Real-Life Application : Examples of Data Modeled Using a Linear Formula

Find an article through newspapers, magazines, professional journals, etc and find at least two examples of data that are best modeled using linear formulae. Describe the importance of each example and why a linear model is appropriate for the data. Note that we are referring to a linear model not simply a time chart where dots