Proof of limits as one function goes to zero and one is bounded Suppose that $$ \lim_{x\to +\infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Show that $$\lim_{x\to +\infty}f(x)g(x)=0$$
Thanks in advance 

Let $\alpha$ be an upper bound of b such that -$\alpha$ is a lower bound for g. Then alpha is a constant. This means that ($\alpha f_x$) converges to 0. 
For each $\epsilon>0$, choose the m that you'd choose for ($\alpha f_x$). Let x be in $\mathbb R$ and m < x. Then $\left |f_x g_x-0 \right |< \left |\alpha f_x-0 \right|<\epsilon$. Thus ($f_xg_x$) converges to 0 as well. 
Q.E.D
Adam V. Nease
