- #1

diligence

- 144

- 0

Say you want to use a proof by contradiction to prove that a sequence diverges. So you assume that x(n)-----> L , and try to find a real number, call it M, such that |x(n) - L| can never get smaller than M, thus arriving at a contradiction.

My question is: can M be of the form that it includes previous terms in the sequence? For example, is it sufficient to say:

|x(n) - L| >= |(4 - L)/4x(n-1)| = M, for all natural n.

Notice that the number I'm trying to use as a lower bound for the distance between x(n) and the supposed limit includes the previous term in the sequence as an inverse factor.

I'm having trouble wrapping my head around this because my intuition tells me that this is insufficient, since you could just go farther out in the sequence to eventually get smaller than the original term's M. However, every term in the sequence will have it's own M which depends on the previous term in the sequence. I'm having trouble determining what this means.

Any help please?