Show that any sequence $(u_{n})$ must tends to infinity as $n→∞$

Question

The motiation to this question can be found in About the solution of a difference equation

My question is: Show that any sequence $(u_{n})$ verifying the equation in the above question must tends to infinity as $n→∞$. The expression for $(u_{n})$ is given by:

$$u_{n}=r^{n^2}\left(\sum_{m=1}^{n}\frac{u_1-ru_0+2(m-1)}{r^{m^2}}+u_0\right)$$