I have to learn a long list of proofs. In order to understand them I try to reformulate them and make them easier for me to grasp. In the process of doing so I often make them incorrect. I wonder if the following is correct?
Prove that: $$\lim_{x \to \infty} f(x)g(x) = 0$$ if $$\lim_{x \to \infty} f(x) = 0$$ and $g(x)$ is bounded.
Proof: Since $g(x)$ is bounded $\exists M: |g(x)|< M$, and since $\lim_{x \to \infty} f(x)$ we have $$\forall \varepsilon > 0, \exists \delta: x > \delta \Rightarrow |f(x)| < \varepsilon.$$ We then get $$|g(x)|< M \Rightarrow |f(x)g(x)| < |f(x)\cdot M|$$ and combine this with the limit definition $$\forall \varepsilon > 0, \exists \delta: x > \delta \Rightarrow |f(x)| < \varepsilon \Leftrightarrow \varepsilon \cdot M > |f(x) \cdot M| > |f(x) g(x)|$$ q.e.d.
Almost. Here are some things that I would modify:
Let $\varepsilon >0$. Since ${\displaystyle \lim_{x\rightarrow\infty}}f(x)=0$, there exists $\delta >0$ such that $|f(x)|<\varepsilon/M$ whenever $x>\delta$. Assume $x>\delta$. Then $|f(x)g(x)|=|f(x)||g(x)|< |f(x)|M<(\varepsilon/M)M=\varepsilon$.