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Let $\{a_t\}$ be a sequence of real numbers that converges to some limit $a$ as $t\rightarrow\infty$. Now let $f$ and $g$ be any functions $\mathbb{R}\rightarrow\mathbb{R}$, and define the following sequence: \begin{align*} b_t &= \frac{\sum_{s=1}^{t} f(a_s)g(a_s) }{\sum_{s=1}^{t} f(a_s)} \end{align*} Does the sequence $\{b_t\}$ converge? I want to say that it does, and its limit is equal to $g(a)$, but I'm not sure if this is in fact correct, and how I'd go about proving it if it is.

[edit in response to Andre and Taro] Thanks for the responses -- you're right: I should've included tighter conditions. Say we add in the requirement that $a_t \in [0, c]$ for some positive real number $c$, and that we have the limit $a\in (0, c)$. Also $f$ and $g$ are both continuous on $(0, c)$, and $f > 0$ on $[0,c]$ (so Taro's example is excluded).

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'Any" functions is awfully broad. If $g$ is not continuous at $a$, one cannot expect any connection between the limit, if it exists, and $g(a)$, – André Nicolas Mar 21 '13 at 2:58

It's false.

Suppose $a_t = 1/t$, $f(x) = 1$, and $g(x) = 1/x$ for $x > 0$. Then $$ b_t = \frac{\sum_{s = 1}^t s}{\sum_{s = 1}^t 1} = \frac{t(t + 1)/2}{t} = \frac{t + 1}{2} \to \infty$$ as $t \to \infty$.

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