# Elementary convergence

by meemoe_uk
Tags: convergence, elementary
 P: 118 Hi everyone, I'm doing a course which contains foundation work on convergence. I was suprised to see the book I am using uses phrases such as.... " This sequence clearly doesnt converge " for sequences such as 2,0,2,0,2,0,2,0..... I was expecting it to say something like " By theorem 4.5, this sequence doesnt converge " I wouldnt feel comfortable writing " This sequence clearly doesnt converge " if, in an exam, I got a question which said " Prove that 2,0,2,0,2,0 doesnt converge ". Can anyone point me to basic theorems on convergence which are used to tackle simple questions like this?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,288 I don't (like your book) see any reason to appeal to a "theorem". When your text says "clearly" what it means is that it follows directly from the definition. A sequence of numbers {an} converges to a limit, L, if, by going far enough on the sequence all the numbers past that point are arbitrarily close to L. Formally: for any [epsilon]>0, there exist an integer N such that if n> N, |an-L|< [epsilon]. ("n> N" is "far enough on the sequence", "|an-L|" measures the distance from an to L and "< [epsilon]" is the "arbitrarily close" part.) Take [epsilon]= 1/2. Two consecutive terms are 2 and 0 and they can't both be with distance 1/2 of anything.
 P: 118 Well, if I wanted to decide if 2,0,2,0,2 converged then I wouldnt need to study a bunch of theorems to convince myself it didnt, because it is clear to my intuition that it doesnt. But I cant just write that in an exam. Since I started this maths degree, there's been loads of questions I've been confronted with where the answers are so blatently obvious that I feel like writing " Because it just bloody is! OK? ", but you cant write that. You've gotta apply the fundamental theorems. Have you attempted a direct proof in what you've written? Looks OK, part from the last line. If there's no theorem to fall back on, then I spose I'd have to construct one myself, maybe with induction method. I like the way you write "theorem", like you think it's a word I've made up.
Emeritus
PF Gold
P: 16,100
Elementary convergence

 Have you attempted a direct proof in what you've written?
Do an indirect proof.

Suppose both 2 and 0 are within distance 1/2 of L.
IOW |2 - L| < 1/2 and |L - 0| < 1/2
Now apply the triangle inequality:
2 = |2 - 0| = |2 - L + L - 0| < |2 - L| + |L - 0| < 1/2 + 1/2 = 1
So 2 < 1
So the supposition was false, and both 2 and 0 cannot be within distance 1/2 from the same number.

(the triangle inequality is one of your best friends when working with &epsilon;-&delta; proofs)
 P: 118 Thanks hurkyl

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