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 doesn`t 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 doesn`t converge " I wouldn`t feel comfortable writing " This sequence clearly doesn`t converge " if, in an exam, I got a question which said " Prove that 2,0,2,0,2,0 doesn`t converge ". Can anyone point me to basic theorems on convergence which are used to tackle simple questions like this?
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 {a_{n}} 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, |a_{n}-L|< [epsilon]. ("n> N" is "far enough on the sequence", "|a_{n}-L|" measures the distance from a_{n} 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.
Well, if I wanted to decide if 2,0,2,0,2 converged then I wouldn`t need to study a bunch of theorems to convince myself it didn`t, because it is clear to my intuition that it doesn`t. But I can`t 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 can`t 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.
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 ε-δ proofs)