# Convergence

• Oct 19th 2009, 05:58 AM
aman_cc
Convergence
$[a_n]$ and $[b_n]$ be two sequences

Let Limit n->oo ( $a_n-b_n$) = 0 (i.e Limit n tends to infinity, an=bn)

Is Limit n->oo ( $a_n^2 -b_n^2)$ =0?

(I'm trying to prove that there might exist nested rational interval with no rational point - I need the above for that. Essentially I have setup $a_n^2<2$ and $b_n^2>2$, thus will show that if x belongs to this interval then $x^2=2$ => $x$ is not rational)

Thanks
• Oct 19th 2009, 06:38 AM
Plato
Quote:

Originally Posted by aman_cc
$[a_n]$ and $[b_n]$ be two sequences
Let Limit n->oo ( $a_n-b_n$) = 0 (i.e Limit n tends to infinity, an=bn)
Is Limit n->oo ( $a_n^2 -b_n^2)$ =0?

If you know that the two sequences are both bounded, then yes the result follows.
From what you said you are doing, it seems the sequences would be bounded.

Just notice that $|a^2-b^2|=|a-b||a+b|.$
• Oct 19th 2009, 06:58 AM
aman_cc
Quote:

Originally Posted by Plato
If you know that the two sequences are both bounded, then yes the result follows.
From what you said you are doing, it seems the sequences would be bounded.

Just notice that $|a^2-b^2|=|a-b||a+b|.$

Thanks Plato. Indeed they are bounded.