I have a basic question: if you have a linear combination of two vectors u, v where u = $ \begin{bmatrix} a_1 \\ \vdots \\ a_n \ \end{bmatrix} $ and v = $ \begin{bmatrix} w_1 \\ \vdots \\ w_n \ \end{bmatrix} $ and where $\ w_1+w_2+\cdots+w_n = 1 $ and $\ w_1, w_2, \ldots ,w_n $ are all positive real numbers and $\ a_1, a_2, \ldots ,a_n $ are all positive integers.
My question: Does $ u \cdot v $ converge to say a real number A no matter the size of $ n $?