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Let f and g be real-valued functions defined on (a,infinity). Suppose that lim{x to infinity} f(x)...

Let f and g be real-valued functions defined on (a,infinity). Suppose that 
lim{x to infinity} f(x) = L and lim{x to infinity} g(x) = M, where L, M are both real. Prove that lim{x to infinity} fg(x) = LM.

(hint: assume 0 < epsilon < 1. Choose a value N1 such that |f(x)-L| < epsilon / [2(|M| + 1)] for all x > N1. Use the triangle inequality with |f(x) g(x) - LM| = |f(x) g(x) - Lg(x) + Lg(x) - LM| .)

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