- #1

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## Homework Statement

For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##.

Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_n) = \lim_{n\to\infty}a_n##, for every convergent sequence ##\{a_n\}_{n=1}^\infty\subset\mathbb{R}## if and only if, for each ##m\in\mathbb{N}##, ##\lim_{n\to\infty}b_{n,m}=0##.

## Homework Equations

## The Attempt at a Solution

I really need at least one hint for this one. Which direction of the proof should I start with? Which one is easier?

My idea for the <---- direction is that since for each ##m\in\mathbb{N}##, ##\lim_{n\to\infty}b_{n,m}=0##, we see that ## \lim_{n\to\infty}b_{n,1}\cdot a_1+\lim_{n\to\infty}b_{n,2}\cdot a_2+\cdots+\lim_{n\to\infty}b_{n,n}\cdot a_n) = 0 + 0 + \cdots + 0##... But I'm not sure where this gets me.