Let $\{a_n\}$ be a sequence of complex numbers and $\{A_n\}$ be a non-decreasing sequence of positive numbers, tending to infinity. Let $\{X_n\}$ be a integrable, centered sequence of i.i.d. random variables. Put $Y_n=X_n \mathbf{1}_{ \{ |X_n|\le A_n/|a_n| \} } $. We have $\lim\limits_{n\to\infty} \mathbb{E} (Y_n)=0$ and $$\sup\limits_{n\ge 1} {\dfrac{\sum\nolimits_{k=1}^n {|a_k|}}{A_n}} <\infty.$$ Proof that $\lim\limits_{n\to\infty} \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{n} { \mathbb{E} (a_k Y_k) } =0$.
In my opinion, since $ \lim\limits_{n\to\infty} \mathbb{E} (Y_n) =0 $ so for all $\varepsilon>0$, exists $N\in \mathbb{N}^*$ such that $| \mathbb{E} (Y_n) |\le \varepsilon$ for all $n\ge N$. Hence \begin{align*} \left| \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{n} { \mathbb{E} (a_k Y_k) } \right| %& \le \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{n} {\left| \mathbb{E} (a_k Y_k) \right| } &\le \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{N-1} {\left| \mathbb{E} (a_k Y_k) \right| } + \dfrac{1}{A_n}\displaystyle\sum_{k=N}^{n} |a_k|. {\left| \mathbb{E} ( Y_k) \right| } \\ & \le \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{N-1} {\left| \mathbb{E} (a_k Y_k) \right| } + \varepsilon.\dfrac{1}{A_n}\displaystyle\sum_{k=N}^{n} |a_k| \\ & \le \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{N-1} {\left| \mathbb{E} (a_k Y_k) \right| } + \varepsilon.M, \end{align*} with $M=\sup\limits_{n\ge 1} {\dfrac{\sum\nolimits_{k=1}^n {|a_k|}}{A_n}}$. Take $n\to\infty$ and $\varepsilon \to 0$ we obtain $\lim\limits_{n\to\infty} \dfrac{1}{A_n}\displaystyle\sum_{k=1}^{n} { \mathbb{E} (a_k Y_k) } =0$.
Is my above proof right or wrong? Can I replace condition: $\sup\limits_{n\ge 1} {\dfrac{\sum\nolimits_{k=1}^n {|a_k|}}{A_n}} <\infty$ by $\lim\limits_{n\to \infty} {\dfrac{\sum\nolimits_{k=1}^n {|a_k|}}{A_n}} <\infty$?