Let $\{a_n\}\mid\sum\sqrt{a_n}

Question

Let $\{a_n\}\mid\sum\sqrt{a_n}<\infty$ and $\{b_n\}\in\mathbb R$. Now:$$E=\left\{\frac{a_n}{\left|x-b_n\right|}<\infty\right\}$$Prove that $\mathbb R-E=\emptyset$.

anonymous · Answer

@Zarkon @FoolForMath @KingGeorge @TuringTest :3 don't mind if I drag you in.

kinggeorge · Answer

Just to be clear, $\emptyset$ is denoting the empty set correct?

anonymous · Answer

Yes.

anonymous · Answer

Made a correction. :\ Minor detail, but important.

kinggeorge · Answer

I suppose we also have to consider complex sequences of $\{a_n\}$?

anonymous · Answer

I think so.

kinggeorge · Answer

In that case, I have no idea whatsoever.

anonymous · Answer

If we were to restrict $\{a_n\}\in\mathbb R$, what would you do? I still don't know what I'd do about this.

kinggeorge · Answer

Even then, I'm still not sure where to go. If we were to restrict everything to the positive reals, I might be able to get somewhere, but I'm still not sure where to start.

anonymous · Answer

Well, thanks for the attempt. :)

anonymous · Answer

Are the sequences $$ a_n$$ and $$b_n$$ fixed and we consider the set E of x in R so the
the inequality is satisfied?

anonymous · Answer

@eliassaab Yes.

anonymous · Answer

Take
$$a_n =\frac 1 {n^4}\
b_n =0,\text { for all  }n
$$
The number x=0 does not belong to E. So R-E is not the empty set.

anonymous · Answer

Alright, cheers.