1. ## Convergence

$\displaystyle [a_n]$ and $\displaystyle [b_n]$ be two sequences

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

Is Limit n->oo ($\displaystyle a_n^2 -b_n^2)$ =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 $\displaystyle a_n^2<2$ and $\displaystyle b_n^2>2$, thus will show that if x belongs to this interval then $\displaystyle x^2=2$ => $\displaystyle x$ is not rational)

Thanks

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

3. 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 $\displaystyle |a^2-b^2|=|a-b||a+b|.$
Thanks Plato. Indeed they are bounded.