# Elementary convergence

#### meemoe_uk

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?

#### HallsofIvy

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.

#### meemoe_uk

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.

#### Hurkyl

Staff Emeritus
Gold Member
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)

#### meemoe_uk

Thanks hurkyl

"Elementary convergence"

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