- #1
TeenieBopper
- 29
- 0
Homework Statement
Use the fact that an= a + (an - a) and bn= b + (bn - b) to establish the equality an*bn - ab = (an-a)(bn-b)+b(an-a)+a(bn-b).
Then, use this equality to prove that the sequence {an*bn} converges to ab.
Homework Equations
Definition of convergence: |an*bn - ab| < ε
The Attempt at a Solution
The first part was easy; just basic algebra. I'm stuck on the last part. I'm not sure where to begin. I tried expanding out the right side, hoping to find something I could use the triangle inequality on. I ended up with
an*bn - ab= an*bn -b*an -a*bn +ab+b*an-ab+a*bn-ab
= an*bn+a*bn-ab
= bn(an+a)-ab
I don't think I can do anything with that. Any suggestions where I can go from here? Am I even starting in the right place?