Proper punctuation of cases in statement of Lemma

Question

I found a very similar question asked on the TeX StackExchange here but the answer was not as definitive as I hoped for.

My question is in regard to how one properly punctuates the following Lemma. Should a comma be used at the end of each case? If not, what would be the proper way to punctuate the cases environments?

Lemma 6. Under the specified parameters constraints in Definition 3 \begin{array}{*3{>{\displaystyle}l}} &\lim_{z\to 0}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega+\nu)^{(n)}, &\nu<0,\\ (\omega)^{(n)}, &\nu>0, \end{cases}\\[1em] &\lim_{z\to\infty}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega-1)^{(n)}, &\nu<0,\\ (\omega-1+\nu)^{(n)}, &\nu>0. \end{cases} \end{array}

Answer 1

There are not really any hard rules for these situations that I am aware of, but I would pick either

Option One:\begin{array}{*3{>{\displaystyle}l}} &\lim_{z\to 0}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega+\nu)^{(n)}, &\nu<0\\ (\omega)^{(n)}, &\nu>0, \end{cases}\\[1em] &\lim_{z\to\infty}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega-1)^{(n)}, &\nu<0\\ (\omega-1+\nu)^{(n)}, &\nu>0. \end{cases} \end{array}

or

Option Two:\begin{array}{*3{>{\displaystyle}l}} &\lim_{z\to 0}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega+\nu)^{(n)}, &\nu<0\\ (\omega)^{(n)}, &\nu>0 \end{cases}\\[1em] &\lim_{z\to\infty}\tilde g_{n,\omega}(z,\nu)% &=% \begin{cases} (\omega-1)^{(n)}, &\nu<0\\ (\omega-1+\nu)^{(n)}, &\nu>0. \end{cases} \end{array}

If you are using commas instead of, for example, writing $$(\omega-1)^{(n)} \ \text{ for }\ \nu<0$$ then you definitely do want the first set of commas and you definitely don't want commas after every inequality, but you should have the period at the end, and it's a matter of taste whether the comma at the end of the first limit is necessary.