Some bounded sequences that always converge

Let $\alpha\in\mathbb{C}$ be such that $\vert \alpha\vert\neq1$ and $u$ be a complex sequence such that $u_{n+1}-\alpha\,u_n\,\underset{n\to +\infty}{\longrightarrow} 0$.

1. We assume furthermore that $u$ is bounded. Show that $u$ converges.
2. For which $\alpha$ can the result of 1) be generalized if $u$ is not bounded ?
3. Show that the sequence $(e^{in\theta})_{n\in\N}$ does not converge for all $\theta\in\mathbb{R}\backslash 2\pi\mathbb{Z}$. Conclude for the case $\vert\alpha\vert=1$.