how can i prove that $x_n$ convergences?

Question

given that:

$$\left(\limsup\limits_{n\rightarrow\infty}\left(\frac{1}{x_n}\right)\right)\cdot \left(\limsup\limits_{n\rightarrow\infty}(x_n)\right)=1$$

How can i prove that $x_n$ convergences?

I don't have ideas..

Answer 1

See my comment above, we need the further hypothesis $x_n >0$ (eventually).

Call $L = \limsup_{n \to \infty} x_n$. By your condition, and our hypothesis we have clearly $0 <L < \infty$. In particular $L^{-1} = \limsup_{n \to \infty} x_n^{-1}$.

Now, it comes the key point: you have to show, only using definitions, that $$ \limsup_{n \to \infty} \ x_n^{-1} = \left( \liminf_{n \to \infty} \ x_n \right)^{-1}$$ I leave this exercize to you, since the solution depends on what definition of $\limsup$ you are familiar with. Hence, we have that $$\liminf_{n \to \infty} x_n=L = \limsup_{n \to \infty} x_n$$ and this concludes the proof.