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## Homework Statement

show that if F:(a,∞) -->R is such that lim xF(x) = L, x --> ∞, where L is in R, then lim F(x) = 0, x --> ∞.

## Homework Equations

## The Attempt at a Solution

Let F:(a,∞) →R is such that lim xF(x) = L, x → infinity, where L is in R. Then there exists an α> 0 where given ε, there exist k(ε) for all x > k then ε > max{1 , ([L]+1)/x} Therefore [xF(x) - L] < 1 whenever x > α. Therefore [F(x)] < ([L]+1)/x. Thus [F(x)-0] < ε Then lim F(x) =0 as x → ∞. This is what I have but it doesn't look right to me.

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