Let $\{u_n\}$ be a sequence of nonnegative numbers satisfying the condition $$ \tag{1} u_{n+1}\leq (1-\alpha_n)u_n+\beta_n \quad \forall n\in\mathbb{N}, $$ where $\{\alpha_n\}$ and $\{\beta_n\}$ are sequences of real numbers such that
$\tag{2}\displaystyle\lim_{n\rightarrow\infty}\alpha_n=0$
$\tag{3}\displaystyle\sum_{n=0}^{\infty}\alpha_n=\infty$
$\tag{4} \displaystyle\sum_{n=0}^{\infty}\beta_n<\infty$
Prove that $$\displaystyle\lim_{n\rightarrow\infty}u_n=0$$