True or false? If true, justify. If false, give counterexample. If $f,g : \mathbb{R} \to \mathbb{R}$ are functions such that $f$ is bounded and $\lim_{x \to +\infty} g(x) = +\infty$, then $\lim_{x \to +\infty} f(x) + g(x) = +\infty$.
I could not think of a counterexample, so I assumed it is true. I am trying to get used to proving these kinds of statements, so I tried the following:
$f$ is bounded, so $\lim_{x \to +\infty} f(x)$ exists and is finite. $g$ increases without bound as $x \to +\infty$. Therefore, $\lim_{x \to +\infty} f(x) + g(x) = +\infty$.
Is it incomplete? Is it rigorous? Is it correct? Any help is appreciated.
Edit: Just noticed that "$f$ is bounded, so $\lim_{x \to +\infty} f(x)$ exists and is finite" is false. For example, $f(x)=\sin{x}$.
Since $f$ is bounded, there exists $M$ for which $|f(x)| < M$ for any $x$. Hence $\lim_{x \rightarrow+\infty}f(x) + g(x) \geq \lim_{x \rightarrow+\infty} g(x) - M = +\infty$.