Let $0\le p\le \infty$ and $q$ be conjugate exponents. $f\in L^p(\mathbb R^n)$ and $g\in L^q(\mathbb R^n)$. To prove $f*g$ is bounded, do I need to prove $f*g$ is bounded from above and from below?
My attempt: \begin{align*} (f*g)(x)=\int_{\mathbb R^n} f(x-t)g(t)dt &\le \int_{\mathbb R^n} |f(x-t)g(t)|dt\\ &\le \left(\int_{\mathbb R^n} |f(x-t)|^pdt\right)^{1/p}\left(\int_{\mathbb R^n} |g(t)|^q dt\right)^{1/q}\\ &=\left(-\int_{\mathbb R^n} |f(x-t)|^p d(x-t) \right)^{1/p}\left(\int_{\mathbb R^n} |g(t)|^q dt\right)^{1/q}\\ &=(-1)^{1/p}||f||_p||g||_q\\ &<+\infty. \end{align*} Is the third line correct?
Should I also prove $f*g$ is bounded from below? How to proceed?
The third line of your proof is indeed correct, as pointed out by @Alonso Delfin in comment above that Lebesgue measure is translation invariant.
Your proof has also shown correctly that $f*g$ is bounded.