Sequences Definition and 576 Threads

Consider the sequence \{ a_{n} \} If |a_{n+1}| > |a_{n}| Prove that \lim_{n→∞} a_{n} ≠ 0 The problem is part of a proof I am trying to understand, but I don't understand this particular step in the proof. Any ideas on how I might grasp this step? BiP

How can we prove this statement? It two subsequences (a2n) and (a2n-1) converge to the same value,then (an)) converges to that value also.

How can you identify the class of all sequences that converge in the cofinite topology and to what they converge to? I get the idea that any sequence that doesn't oscillate between two numbers can converge to something in the cofinite topology. Considering a constant sequence converges to the...

I have a question; help is welcome. Let sn be a linear, non-homogeneous recurrence sequence, and let hn be a corresponding homogeneous sequence (with initial values to be determined). As it turns out, the offset between the two (sn - hn) is given by the steady state value of sn, if the...

Homework Statement Let (an) be an increasing sequence. Let (bn) be a decreasing sequence and assume that an≤bn for all n. Show that the lim(an)≤lim(bn). Homework Equations I will show that my sequence are bounded above and below, respectively. Thereby forcing the monotone sequence to...

i have deleted the question as it appeared no one would answer it

Homework Statement Use the fact that a_n=a+(a_n-a) and b_n=b+(b_n-b) to establish the equality (a_n)(b_n)-ab=(a_n-a)(b_n)+b(a_n-a)+a(b_n-b) Then use this equality to give a different proof of part (d) of theorem 2.7. Homework Equations The theorem it is citing is: The sequence...

I know in general how to prove if sequence is null or not . But here is my confusion - method in the textbook I am reading - asks that to prove that any sequence is null we must show that for each ε>0, there is an integer N such that modulus of the given sequence is < ε, for all n>N now I...

Homework Statement The problem is: Show that an \rightarrow \infty iff for all \Delta > 0, \existsN such that n \geq N \Rightarrow an \rightarrow \infty Homework Equations Not sure if there are any The Attempt at a Solution I can't really think of anything to do here because I have...

considering divergence of a sequence in the reals, a_{n}, if such a sequence → +∞ as → n, then I would like to know what type of sequence this reuqires. (excluding divergence to -∞ for now) so a_n → +∞ iif: \forall M \exists N, \forall n\geqN \Rightarrow a_n \geq M . So is the above...

Hello all. I have to present a proof to our Intro to Topology class and I just wanted to make sure I did it right (before I look like a fool up there). Proposition Let c be in ℝ such that c≠0. Prove that if {an} converges to a in the standard topology, denoted by τs, then {can}...

Now I know how this works- but I came across this example and even though I know the answer- the simplification given in the explanation doesn't make sense to me. the squence is an= {5n/n!} now applying an+1 and dividing an+1/an the book indicates = 5/n+1 this is what I don't get how (5n+1...

Homework Statement Why is it that continuous functions do not necessarily preserve cauchy sequences. Homework Equations Epsilon delta definition of continuity Sequential Characterisation of continuity The Attempt at a Solution I can't see why the proof that uniformly continuous...

In Dummit and Foote, a short exact sequence of R-modules 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 (\psi:A \rightarrow B and \phi:B \rightarrow C) is said to split if there is an R-module complement to \psi(A) in B. The authors are not really clear on what the phrase "an R-module...

Sequences and Series Problem. Help! Pleeaassee! Homework Statement I've attached the problem and my work. I'm supposed to express the sum as a fraction of numbers in lowest terms. The original statement was: 2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5) + ... + 2/(100*101*102) and the answer is 2575/5151...

Choose two whole numbers, say 2 and 1. Add them and yield 3; multiply them and yield 2. Repeat using those new numbers. 3+2=5; 3x2=6 5+6=11; 5x6=30 11+30=41; 11x30=330 41+330=371; 41x330=13530 etc. Have such sequences been explored before? Their generation is relatively...

Homework Statement I would like to show that a weakly convergent sequence is necessarily bounded. The Attempt at a Solution I would like to conclude that if I consider a sequence {Jx_k} in X''. Then for each x' in X' we have that \sup|Jx_k(x')| over all k is finite. I am not sure why...

Hi All, So here's my question: Suppose we have two sets A and B, then A \setminus B denotes their set-difference. Does there exist an equivalent operator for the case where A and B are not sets, but sequences? Otherwise, is there an operator to convert a sequence into a set...

Homework Statement I'm supposed to answer true or false on whether or not the sequence ((-1)^n * n) tends toward both ±∞ Homework Equations Uniqueness of Limits The Attempt at a Solution I did prove it another way, but I would think that uniqueness of limits (as a definition...

Homework Statement If I have a constant sequence a,a,a,a,a,a, Is that monotonic? because I have conflicting definitions some say that it has to increase or decrease, but some say it just has to not increase or not decrease. And also does every bounded sequence have monotonic sub...

Homework Statement Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norms are defined on l_infty(R) . If they are, verify by axioms. If not, provide counter example. Homework Equations x in l_infty(R); x=(x_n), (i) || ||_# defined by...

Why are Cauchy sequences important? Is there only purpose to test convergence of sequences or do they have other applications? Is there anything tangible about Cauchy sequences

Homework Statement let {xi} be a sequence of distinct elements in a metric space, and suppose that xi→x. Let f be a one-to-one map of the set of xis into itself. prove that f(xi)→x Homework Equations by convergence of xi, i know that for all ε>0, there exists some n0 such that if i≥n0...

Hello all, I am having trouble with a homework problem. The problem is as such: Let a = {zn = (xn,yn) be a subset of ℝ2 and zn be a sequence in ℝ2 such that xn ≠ xm and yn ≠ ym for n≠m. Let Ax and Ay be the projections onto the x and y-axis (i.e. Ax = {xn} and Ay = {yn}. Assume that the...

Just a little help understanding results obtained. I have found the closed form of a sequence, but am a little unsure if there is a right way or can select either way of using the terms to create the explicit formula. I have found the common difference from the terms, which is 1.2, in my...

Here is the proof provided in my textbook that I don't really understand. Suppose that x' and x'' are both limits of (xn). For each ε > 0 there must exist K' such that | xn - x' | < ε/2 for all n ≥ K', and there exists K'' such that | xn - x'' | < ε/2 for all n ≥ K''. We let K be the larger...

I have a linear recurrence sequence, 3, -1.5, 0.75, -375 x = a a = 3 x2, = -1.5, x3, = 0.75, x4 = -375... x2 = rx1+d x3 = rx2+d -1.5 = 3r + d 0.75 = -1.5 + d -1.5 - 0.75 = (3r + d) - (-1.5 + d) r = - 0.5 Sub in equation (2) d = -1.5 - 3r = -1.5 - 3(-0.5) d = 0 x4 = -0.5 x 0.75 + 0 =...

Hello all, I am having trouble with a convergent series problem. The problem statement: Let f:ℝ→ℝ be a function such that there exists a constant 0<c<1 for which: |f(x)-f(y)| ≤c|x-y| for every x,y in ℝ. Prove that there exists a unique a in ℝ such that f(a) = a. There is a provided hint...

Homework Statement Consider P[0,1] the linear space of C[0,1] consisting of all polynomials. Show that the sequence {pn} where pn(t)=tn has the property that its coefficient sequences converge but the sequence {pn} does not converge in (P[0,1], ∞-norm). Homework Equations Observation...

Homework Statement a) Let {s_{n}} and {t_{n}} be two sequences converging to s and t. Suppose that s_{n} < (1+\frac{1}{n})t_{n} Show that s \leqt. b) Let f, g be continuous functions in the interval [a, b]. If f(x)>g(x) for all x\in[a, b], then show that there exists a positive real z>1 such...

Homework Statement Give an example to show that it is not necessarily true that the product of two eventually increasing sequences is eventually increasing. Homework Equations a sequence is eventually increasing if for N\in natural numbers, a_{n+1} \geqa_{n} for all n>N. The Attempt...

As usual, I typed up the problem and my attempt in LaTeX: Maybe I'm not applying the MVT correctly, but my result does not seem to help me solve the problem in anyway. What are your thoughts?

As usually, I type the problem and my attempt at the solution in LaTeX. Ok, so for the last part (c), I obviously have the diagram down, now I just have to construct the nested sequence of functions that converges at every point in A. I drew a diagram to help illustrate the idea...

Hello, below I have the problem and solution typed in Latex. For the first part, I just want someone to verify if I am correct. For the second part, I need guidance in the right direction

So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such. I've come to understand that this property follows...

Homework Statement Let 1\leq p,q that satisfy p+q=pq and x\in\ell_{p},\, y\in\ell_{q}. Then \begin{align} \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{\infty}\left\vert...

Homework Statement prove that if k>1 then kn→∞ an n→∞ there is a hint given. (hint:let k=1+t where t>0 and use the fact that (1+t)n>1+nt) (ii) prove that if 0<k<1 then kn→0 as n→∞ Homework Equations The Attempt at a Solution if k>1 then kn+1-kn=kn(k-1) (ii) 1/k>1 then1/kn→∞

Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...

Homework Statement Show that A, the set of all increasing sequences of natural numbers is uncountable Homework Equations I know that the natural numbers themselves are countable. The Attempt at a Solution I am thinking of using some sort of diagonal argument to prove this.

if lim(n→∞)an=0 then prove lim(n→∞)1/an=0 how do i do this, i know how to proove it geometrically, but how do you write the proof using ε and \delta Give a counter example to show that the converse is false.

given closed subsets, A and B, of R^d with A bounded prove the equivalence of: 1) There exists a pair of sequences x_n in A and y_n in B such that |x_n - y_n| -> 0 as n -> infinity 2) A intersection B is non empty. I have attempted this question but am a bit stuck on proving 1 implies 2...

Homework Statement Let {xn} be a sequence of real numbers. Let E denote the set of all numbers z that have the property that there exists a subsequence {xnk} convergent to z. Show that E is closed. Homework Equations A closed set must contain all of its accumulation points. Sets with no...

If f is continuous function and (x_n) is a sequence then x_n \to x \implies f(x_n) \to f(x) The converse f(x_n) \to f(x) \implies x_n \to x in general isn't true but why is it true, for example, if f is arctan?

Hello everyone! My question is twofold. Firstly, how do I solve for term numbers in a geometric sequence and secondly, how do I algebraically solve for variables that are exponents? Homework Statement Given the following geometric sequences, determine the number of terms, n. t1=5 r...

Homework Statement Find the sum of the products of every pair of the first 'n' natural numbers. Homework Equations sigma n^2 = n(n+1)(2n+1)/6 The Attempt at a Solution S=1.2 + 1.3 + 1.4 ...+ 2.3 + 2.4 ...n-1(n) i can't figure out how to proceed ..

Homework Statement (excuse lack of latex) show that if SUM(zn)= S and SUM(wn= T, then SUM(zn + wn) = S + THomework Equations The Attempt at a Solution so if I'm doing this right, this is pretty easy, i think. they want me to use a theorem that says if zn=xn +iyn, and SUM(zn)= S, where S = X +...

Let A be a non-empty subset of the complex plane and let b ∈ ℂ be an arbitrary point not in A. Now define d(A,b) := inf{|z-b| : z ∈ A}. Show that if A is closed, then there is an a ∈ A such that d(A,b) = |a-b|. Ok so basically what I did was begin by choosing some arbitrary element of A and...

I know that Q (rational numbers) are countable and that the finite cartesian of Q with itself, Q^n is countable but is it true that the countably infinite cartesian product of Q with itself is countable? The set of all rational sequences are isomorphic to Q^∞ (here I am saying Q^∞ is the...

Homework Statement a) Prove that \ell_\infty \mathbb({R}) is a subspace of \ell \mathbb({R}) b) Show that \left \| \right \|_\infty is a norm on \ell_\infty (\mathbb{R}) The Attempt at a Solution For a) I guess we have to show that \vec{x} + \vec{y} \in \ell_\infty \mathbb({R})...

Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...