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# Real Analysis

### Convergent Series Sum

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

### Real Analysis Summation

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

### 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 - Series Converges

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

### 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 R.show 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.)

### Infinite Series, Convergence and Divergence Test

PLEASE SHOW ALL WORK, STEP-BY-STEP, WITH ALL CORRECT NOTATION. My notation is lousy for some reason the attachment won't come through, so I cut and pasted it below. Determine whether the following Diverge (D), Converge Conditionally (CC), or Converge Absolutely (AC). Give the rationale for each response. Must show all work

### Real analysis

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

### Real Analysis

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

### Real Analysis

Show that the sequence defined by y1=1 and y_n+1=4-1/y_n converges and find the limit

### 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: Bounded Sequence

Let (a_n) be a bounded sequence and assume lim b_n=0.show that lim(a_n b_n)=0.

### 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 of Irrational Numbers

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

### 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: Proof Regarding Bounded Subsets

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.

### 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