Prove: Advanced Calc Proof: ((a(n))+(a(n+1)))/2 Converges to A

[bloynoys]

Homework Statement

If ((a(n))+(a(n+1)))/2 converges to A, then a(n) converges to A. Either prove or disprove this conjecture.

Homework Equations

Normal convergence proof

The Attempt at a Solution

I will prove that ((a(n))+(a(n+1)))/2 converges to A such that for every (epsilon)>0 there exists a positive integer N such that for every n>N abs(a(n)-A)<(epsilon).

Consider (epsilon)> 0 arbitrary.
Because a(n) and a(n+1) will converge to the same number...

That second part is what I am stuck at. I am really good at proving things converge this way with sequences defined but struggle in the abstracts. I struggle with the second and fourth lines of the proof, as the third is just consider n>N arbitrary.

 

[ehild]

Consider the sequence 1, -1, 1, -1, ...that is a(n)=1 if n is odd, a(n)=-1 if n is even. What is the limit of (a(n)+a(n+1))/2?ehild

 

[bloynoys]

Wow, yeah. And I just used that to disprove the question before that. Thank you!

Another related question on proofs:

One is false and one is true (Prove the true, disprove the false)

Conjecture A: If a(n) is a sequence of real numbers then for every positive integer n there exists an M such that abs(a(n)) <= M.

Conjecture B: If a(n) is a sequence of real numbers there exists an M such that every positive integer n abs(a(n)) <= M.

I have been staring at it and I should know it but am struggling to figure it out.

 

[ehild]

Conjecture A: If a(n) is a sequence of real numbers then for every positive integer n there exists an M such that abs(a(n)) <= M.

Conjecture B: If a(n) is a sequence of real numbers there exists an M such that every positive integer n abs(a(n)) <= M.

I think you miss something from sentence B. abs(a(n)) <= M has to be true for every positive integer n.

Think what do the sentences mean? Imagine a simple sequence, 0.1, 0.2, 0.3,... for example. Is A true? Choose an n, for example n=1000. Can you find an M so as |an|<=M?

Can you find an M so as |a(n)|<=M for every n?

ehild

 

[bloynoys]

Thank you! That is what so frustrating about an entirely proof based class at this level. There is so many ways to look at these things and you understand it in class but when you get home to do questions that are slightly different you just sit there, as I know the stuff, but helps to have someone saying consider this... and then connecting the dots.

 

[Deveno]

sometimes it helps to know what is really being said.

conjecture A states that every term of a real sequence is a finite real number.

conjecture B states that every real sequence is bounded.

put this way, it should be clear which one is true, and which one is false.

(and, on a more basic level, why the order of "for all" and "there exists" can't simply be reversed).