# Thread: Real Analysis - Functional Limits

1. ## Real Analysis - Functional Limits

Let g: A --> R and assume that f is a bounded function on A, which is a subset of R (i.e. there exists M > 0 satisfying |f(x)| <= M for all x in A). Show that if lim as x-->c of g(x) = 0, then lim as x-->c of g(x)*f(x) = 0 as well.

2. I had an idea using

$\displaystyle |f(x)g(x)|=|f(x)||g(x)| \leq M|g(x)|$.

Can we take the limit as x-> c of $\displaystyle M|g(x)|$?