Building on my other answer, we can construct a counterexample to the original question.
Let $\{I_n \mid n\geq 3\}$, \{I'_n \mid n\geq 3\}, $\{J_n \mid n\geq 3\}$, and \{J'_n\mid n\geq 3\} be mutually exclusive events satisfying P(I_n) = P(I'_n) = \frac{1}{2n^2}\qquad\text{and}\qquad P(J_n) = P(J'_n)=\frac{1}{2n^4}. Let $X_1$ and $X_2$ be the random variables X_1(\omega) =\begin{cases}1/n & \text{if }\,\omega\in I_n \\ -1/n & \text{if }\,\omega\in I'_n, \\ 0 & \text{otherwise},\end{cases} \qquad\text{and}\qquad X_2(\omega) =\begin{cases}1/n & \text{if }\,\omega\in I_n\cup J_n, \\ -1/n & \text{if }\,\omega\in I'_n\cup J'_n, \\ 0 & \text{otherwise},\end{cases} For each $n$, let $Y_n$ and $Z_n$ be the random variables Y_n(\omega)=\begin{cases}n & \text{if }\,\omega\in I_n \\ -n & \text{if } \,\omega\in I'_n \\ 0 & \text{otherwise}\end{cases} \qquad\text{and}\qquad Z_n(\omega)=\begin{cases}n^2 & \text{if }\,\omega\in J_n \\ -n^2 & \text{if } \,\omega\in J'_n \\ 0 & \text{otherwise}\end{cases} Then the functions $\{Y_n\mid n\geq 3\}$ and $\{Z_n \mid n\geq 3\}$ are orthonormal. Moreover, $Y_n \in A_1$ for each $n$, and $Y_n + \frac{1}{n} Z_n \in A_2$ for each $n$. It follows that $Z_n\in A_1+A_2$ for each $n$. However, the sum $ \sum_{n=3}^\infty \frac{1}{n}Z_n $ does not lie in $A_1+A_2$.