For instance, given a finite dimensional vector space $V$ we want to show that any norms in this $V$ is equivalent. However, all proofs online consider only two norms, mainly $|| . ||$ and $|| . ||_{p}$ where $1<p< \infty $ when trying to show this statement and they say this is due to "transitivity".
Why is this the case? I think we can define infinitely many norms on a finite dimensional vector space, not just those of the form $||.||_{p}$ for all $1<p< \infty$.
If you know there are constants $c>0, C>0$ such that $$c||.|| \le ||.||_p \le C||.||$$ where $||.||$ is an arbitrarily chosen norm, and if you then have to norms $|.|_a, |.|_b$ you then know there are corresponding constants $c_a, C_a, c_b, C_b$, such that $$c_a |.|_a \le ||.||_p \le C_b |.|_b$$ and $$c_b|.|_b \le ||.||_p \le C_a |.|_a$$
which implies that $|.|_a$ and $|.|_b$ are also equivalent.
(Of course, in the initial statement, $||.||$ must not be some fixed norm, but any norm, with $c,C$ then depending on this norm and on $p$. Also the indices $a$, $b$ are not meant to imply a certain definition of the norm like the $p$-norms, but are just meant to give a name to the two norms).