I am currently reading Ruy Exel's Partial Dynamical Systems, Fell Bundles and Applications where he mentions that for every $C^*$-seminorm $p$ on a C* algebra $A$ one has $$p(a)\leq ||a||$$ for all $a \in A$. While this seems familiar, I cannot recall the reason for this.
Thank you
From what I recall, every $C^{\ast}$-seminorm arises from a $\ast$-homomorphism. ie. $\exists$ a $\ast$-homomorphism $\varphi : A\to B$ between $C^{\ast}$-algebras such that $$ p(a) = \|\varphi(a)\| $$ Your result follows from this because a $\ast$-homomorphism is norm-decreasing.
To prove the result I mention, the argument (I think) is as follows: Take $N := p^{-1}(\{0\})$, then $N$ is a self-adjoint ideal of $A$, so consider $$ \tilde{B} := A/N $$ with the norm $\|a+N\| := p(a)$. Then, $\tilde{B}$ is a $\ast$-algebra, so take $B$ to be the completion of $\tilde{B}$. It should not be too hard to check that this norm induces a $C^{\ast}$-norm on $B$, so $B$ is a $C^{\ast}$-algebra, so your homomorphism is merely $$ \varphi : A\to B \text{ given by } a \mapsto a+N $$