Hadamard's example: ill-posedness of the Cauchy problem for Laplace's equation

Show that the function \(u=u(x,y)\) given by \[ u(x,y)=\frac{e^{ny}-e^{-ny}}{2n^2}\sin nx\qquad\text{for }n\in\mathbb{N} \] is the solution on \(D=\{(x,y)\mid x^2+y^2<1\}\) of the Cauchy problem for the Laplace equation \[ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\qquad u(x,0)=0,\qquad\frac{\partial u}{\partial y}(x,0)=\frac{\sin nx}{n}. \]

Show that there exist points \((x,y)\in D\) such that \(\limsup_{n\to\infty}|u(x,y)|=\infty\).