Recently I am reading a document [1]. In this document, Prof. Cook provide a brief proof of $\mathbf{P} \subseteq \mathbf{NP}$, which is only one sentence:

> It is trivial to show that $\mathbf{P} \subseteq \mathbf{NP}$, since for each language $L$ over $\Sigma$, if $L \in \mathbf{P}$ then we can define the polynomial-time checking relation $R \subseteq \Sigma^* \cup \Sigma^*$ by
$$R(w, y) \Longleftrightarrow  w \in L$$
for all $w, y \in \Sigma^*$.

I know the definition of $\mathbf{P}$ and $\mathbf{NP}$, as in [1], but I still can't understand this proof. Could any one explain the proof to me? Even one sentence is good.

By the way, I think $\Sigma^* \cup \Sigma^*$ should be $\Sigma^* \times \Sigma^*$. Am I right?

Reference
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[1] S. Cook, *The P versus NP problem*, [Online] http://www.claymath.org/sites/default/files/pvsnp.pdf.