Thread: Proving Convergence

Let (

an) be a sequence which converges to 0. Use the definition of a convergent sequence to prove that the sequence (an^2) to 0.

Definition: There exists an L that is Real s.t. for every E>0 there exists N that is a positive integer s.t. for every n that is a positive integer n>N implies that |an - L| < E

Originally Posted by ah-bee

Let (

an) be a sequence which converges to 0. Use the definition of a convergent sequence to prove that the sequence (an^2) to 0.

Definition: There exists an L that is Real s.t. for every E>0 there exists N that is a positive integer s.t. for every n that is a positive integer n>N implies that |an - L| < E

Since it is bounded, i.e. there is so that .

This means, for .

i dont understand how works. can u explain to me how it works?

Originally Posted by ah-bee

i dont understand how works. can u explain to me how it works?

There exists so that . Because convergent sequences are bounded.

To prove that we need to show for any where is sufficiently large.

Give me any , then since it means for .

But then if we have .

thanks, i understand it now, so if i were to get a question saying if an converges to k, prove an^2 converges to k^2, i would approach it in the same way?

Originally Posted by ah-bee

thanks, i understand it now, so if i were to get a question saying if an converges to k, prove an^2 converges to k^2, i would approach it in the same way?

Not exactly in the same way. But it should be similar.