1. Sequence-Constant Inequalities

If you have a sequence $\displaystyle x_{n}$ and an inequality
$\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta$

Does that mean that the point to which the sequence converges (if it does) is less than $\displaystyle \beta$ or does that mean $\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta, \forall n$?

2. Originally Posted by miatchguy
If you have a sequence $\displaystyle x_{n}$ and an inequality
$\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta$

Does that mean that the point to which the sequence converges (if it does) is less than $\displaystyle \beta$ or does that mean $\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta, \forall n$?

In a rather unsurprising fashion, $\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta$ means just $\displaystyle (|x_{n}|)^{\frac{1}{2}}<\beta$ . As simply as that.

They could tell you this inequality is true for any n or only for some (one, a few, infinite, all...) n's, but that's something you can't ask here but rather ask the person who gave you the inequality.

Now, if $\displaystyle x_n\xrightarrow [n\to\infty]{}\alpha$ then $\displaystyle |x_n|\xrightarrow [n\to\infty]{}|\alpha|$ , but nothing else can be said about the relation between $\displaystyle \alpha,\,\beta$ until we know more about the given inequality.

Tonio