We know that $R$ is a field with the usual addition and usual multiplication. Is the set of extended real numbers also a field with the same operations? What are the additive inverse and multiplicative inverse of $\infty$ in the extended field?
(We know ($\infty-\infty$) is an indeterminate form, so $-\infty$ can't be additive inverse of $\infty$. And $0$ can't be multiplicative inverse of $\infty$ because we have $c.\infty=0$ if $c=0$.)
Best Answer
I think you've answered your question. In a non trivial field $0 \ne 1$ and $a + q = a \iff q = 0$. So $\infty + 1 \ne \infty$ which... defeats the purpose of the extended reals.
It's not a field. (And you really shouldn't be surprised.)