1. ## visualizing convegence

I am trying to get a good visual understanding of convergence so I can better understand the definition of a Cauchy sequence.
Is the limit aways going to be in the center of the interval, $\displaystyle (A-\epsilon, A+\epsilon)$.
I guess I dont understand why the $\displaystyle \epsilon/2 >$ 0 is used in the definition.

2. ## Re: visualizing convegence

Originally Posted by delgeezee
I am trying to get a good visual understanding of convergence so I can better understand the definition of a Cauchy sequence.
Is the limit aways going to be in the center of the interval, $\displaystyle (A-\epsilon, A+\epsilon)$.
I guess I dont understand why the $\displaystyle \epsilon/2 >$ 0 is used in the definition.
I'm not sure what definition you have. To me a sequence $\{x_n\}$ is a Cauchy sequence if

$\forall \epsilon > 0~~\exists~ m,n \in \mathbb{N} \ni \left|x_n-x_m\right| < \epsilon$

i.e. the elements get arbitrarily close to one another as the sequence progresses.

is this any clearer?

3. ## Re: visualizing convegence

Im sorry, I meant the proof for theorem that says a sequence is cauchy. It uses the triangle inequality. A sequence is cauchy. iff for each$\displaystyle \epsilon$ >0 there is a positive integer N s.t. if m,n $\displaystyle \geq$ N, then $\displaystyle \mid a_{n} - a_{m} \mid > \epsilon}$ .

The absolute values make me visualize only half the interval that contain two points in the sequence.