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
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
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.)
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
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 the sequence defined by y1=1 and y_n+1=4-1/y_n converges and find the limit
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
Let (a_n) be a bounded sequence and assume lim b_n=0.show that lim(a_n b_n)=0.
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
Limit x to 9 Square root of x - 3(3 is not part of the square root)/x-9
Limit x to 3 9-x^2/x-3
If sup A < sup B then show that there exists an element b belong to B that is upper bound for 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
Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a) 0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b) and a_n --> a and b_n -->b Then prove that a_n*b_n -->a*b Q2.Let f: R --> R be monotonically increasing, i.e. f(x_1)<= f(x_2) for x_1< = x_2. Show that f is measurable. Hint: You may extend f to f':[-in