Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges Let $x_n, y_n &gt; 0$. 

I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?
Let $x_n$ be any positive sequence converging to $0$. Let $y_n$ be $1/|\ln x_n|$ if $n$ is even and let $y_n$ be $2/|\ln x_n|$ if $n$ is odd. Note that $y_n$ converges to $0$. Then $x_n^{y_n}=e^{y_n\cdot \ln x_n}$ which is $e^{-1}$ or $e^{-2}$, alternating with $n$, thus not converging.
You can show easily that $x_n^{y_n}$ converges to $l_x^{l_y}$, with $l_x$ the limit of $x_n$ and $l_y$ the limit of $y_n$... 

Just by using the right $\epsilon_{x,y}$ and $N_{x,y}$ in the definition of convergence.