I was reading proofs on some convergence tests, but I'm stuck on a few parts here:

1. Prove every cauchy sequence is bounded.

Proof: Suppose the sequence {An} is cauchy.

Let E = 1, pick N in Natural Numbers such that n >= N and m >= N

We have |An - Am| < 1, further, we have |An-AN| < 1.

This is the part that I don't understand, how do we know that|An - AN| < 1?

2. For a number r such that |r| < 1, then the series sum {from k=0} r^k = 1/(1-r).

Proof: sum {from k=0 to n} r^k = 1 + r + r^2 + r^3 + . . . + r^n = (1 - r^{n+1}) / (1 - r)

Now, how do we know that equals?

Thank you, y'all!

KK