On the convergence of orthogonal series to $+\infty$

For any sequence of numbers $a_n\downarrow0$, $\sum_{n=1}^\infty a_n^2=\infty$, a uniformly bounded orthonormal system of continuous functions $\varphi_n(x)$ which is complete in $L_2(0,1)$, and a sequence of numbers $b_n$ ($0<b_n\leqslant a_n$) are constructed such that $\sum_{n=1}^\infty b_n\varphi_n(x)=\infty$ everywhere on $(0, 1)$.