Proof of a statement about the continuity of polynomials and rational functions

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My notes do not include a proof to the following:

If $p$ is a polynomial, then $p$ is continuous at every point in $R$. If $r = \frac{p}{q}$ is a ratio of two polynomials, then it is continuous at every point in $R$ where $q \not = 0$.

Can I prove this through the Sequential Continuity type theorem wherein I say that $p$ tends to a limit within delta and so the combination of it and $q$ also converges by the algebraic properties of limits?

Any help would be appreciated, thanks!

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