I am currently working on the convergence analysis for a fixed-point iteration and would like to show the iteration w.r.t. the variable $x(t)\in\mathbb{R}$ can converge to $0$, i.e., $\lim_{t\rightarrow\infty}x(t)=0$. The iteration follows the dynamics as follows.
$x(t+1)=(1-\alpha(t))x(t)+(\frac{\alpha(t)}{\sum_{i=0}^t\alpha(i)})$,
where $\alpha(t)$ satisfies $\sum_{t=0}^{\infty}\alpha(t)=\infty$ and $\sum_{t=0}^{\infty}(\alpha(t))^2<\infty$. $x(t)\in\mathbb{R}$ is a scalar with an arbitrary given initial value $x(0)>0$.
It is well noted that when the term $(\frac{\alpha(t)}{\sum_{i=0}^t\alpha(i)})$ in above iteration is replaced by $(\alpha(t))^2$, then the sequence will converge to 0.
However, it seems to be difficult to evaluate when the term $(\frac{\alpha(t)}{\sum_{i=0}^t\alpha(i)})$ is considered.
I have tried lots of different methods but failed. Does anyone have any idea about this problem? Thank you very much!