Why is every positive linear functional bounded in $C^*$-algebras?
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For self-adjoint elements $a$, we have the inequality $-\lVert a\rVert e\le a\le\lVert a\rVert e$, where $e$ is the identity. So if $f$ is a positive linear functional, $-\lVert a\rVert f(e)\le f(a)\le\lVert a\rVert f(e)$ follows; i.e., $\lvert f(a)\rvert\le\lVert a\rVert f(e)$. For non-selfadjoint $a$, write $a=b+ic$ with $b$ and $c$ selfadjoint and use the result just shown.