Let $a_n$ be a convergent sequence who's limit is $L$ (Should also apply to when its limit is infinite). Let $t_n$ be a sequence of positive numbers such that $\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{t_n}} = \infty $. Let $b_n$ be as follows: $${b_n} = {{\sum\limits_{k = 1}^n {{t_n}{a_n}} } \over {\sum\limits_{k = 1}^n {{t_n}} }}$$
Show that $b_n\to L$.
I got a hint that the proof should be similar to the regular Cesaro mean proof, but I wasn't able to pin in down in the finite case, not to mention the infinite one (I wasn't able to proof the infinite case in the regular Cesaro mean theorem either).
Any help would be greatly appreciated.
P.S ~ I'm loving the chance to finally use $\LaTeX$ :)