Proving that $L=\{a^{j}b^{k}\,:\,j,k\geq0\}$ is regular

Question

I have the following language: $L=\{a^{j}b^{k}\,:\,j,k\geq0\}$. I'm trying to prove that it is a regular language. I can use the following theorem:

Let $r$ be a regular expression then $L[r]$ is regular language.

So I can use $r=a^*b^*$ to be that regular expression and then $L[r]$ is regular. But I think that I'm missing something. What explanation should I give for using that exactly $r$. In other words, how a formal proof should look like? I'm taking automata and formal languages this semester and I need to be as formal as possible. Could you please should how should I prove it "formally"?

Answer 1

I don't think you need any clarification, but if you insist, you could write $$ L = \{a^{j}b^{k} \mid j,k \geqslant 0\} = \{a^{j} \mid j \geqslant 0\}\{b^{k} \mid k \geqslant 0\} = a^*b^*. $$