I want to learn the embedding properties of $L^p$ spaces, but the Wiki description is too sophisticated.

In one sentence, if a function $f \in L^p(\Omega)$, then it must be (or, under what conditions) in $L^q(\Omega)$ for any $1 \le p \le q \le \infty$, i.e., $L^p \subseteq L^q$ for $1 \le p \le q \le \infty$, or the opposite? How to show that?

I think $f \in L^p(\Omega)$ has the norm \begin{align} \|f\|_{L^p} = \left(\int_{\Omega} |f(x)|^p dx\right)^\frac{1}{p} < \infty \end{align} and we have $\|f\|_{L^p} \ge \|f\|_{L^q}$(?) for $1 \le p \le q \le \infty$, thus $\|f\|_{L^q} < \infty$ as well.

BTW, I saw someone denotes $L^p$ as superscript while some other denotes $L_p$ as subscript, which is the formal notation for the Lebesgue integrable function spaces?