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 

$$\lim _{ x\to +\infty  } f(x)=0\Rightarrow \quad \forall x\in R,\exists M>0\quad \left| f\left( x \right)  \right| \le M\\ \forall x\in R,\exists \frac { \epsilon  }{ M } >0\quad \left| g\left( x \right)  \right| \le \frac { \epsilon  }{ M } \\ \left| f\left( x \right) g\left( x \right)  \right| \le \left| f\left( x \right)  \right| \left| g\left( x \right)  \right| \le M\cdot \frac { \epsilon  }{ M } =\epsilon $$ which means $$\lim_{x\to +\infty}f(x)g(x)=0$$
