BG-MO 1989 Problem 2

Prove that the sequence $(a_n)$ , where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$ converges, and determine its limit as $n\to\infty$ .