### Real Analysis

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

- Anthropology
- Art, Music, and Creative Writing
- Biology
- Business
- Chemistry
- Computer Science
- Drama, Film, and Mass Communication
- Earth Sciences
- Economics
- Education
- Engineering
- English Language and Literature
- Gender Studies
- Health Sciences
- History
- International Development
- Languages
- Law
- Mathematics
- Philosophy
- Physics
- Political Science
- Psychology
- Religious Studies
- Social Work
- Sociology
- Statistics

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.

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?

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.

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

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.)

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

Prove that the sequence defined by X1=3 and X_n+1=1/(4-X_n) converges

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

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

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

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

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

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

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

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

Proof: Given any two real numbers a<b ,there exists an irrational number t satisfying a<t<b

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

Let A subset or equal of R be bounded above and let c belong to R.Define the sets c+A and cA by c+A={c+a : a belong to A} and cA={ca : a belong to A}. 1-show that sup(c+A)=c+ Sup A 2-if c>=0,show that sup(cA)=cSupA 3-postulate a similar type of statement for sup(cA)for the case c<0.

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

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.

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