Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that $$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$
Let $a_{n}$, $n \geq 1$ be a sequence of real numbers. If $a_{n} \rightarrow a$, show that
$$b_{n} = \frac{a_{1} + 2a_{2} + 3a_{3} + \cdots + na_{n}}{n^{2}} \rightarrow \frac{a}{2}.$$