Is Conway assuming that when the limit is $\infty$, it exists ?

Question

In the book of Complex Analysis by Conway, at page 31-32, it is given that

and in the next page, it is also given that

However, the author uses this proposition for the example:

But, $R$ can be infinite and zero, and in this particular case, the given limit in the proposition is

$$\lim n+1 = \infty :\quad \text{does not exists}.$$

So, how can we use the proposition 1.4 in this case ?

Is the author considering $\infty$ as a possible limit value - i.e is he assuming the limit exists when the limit is $\infty$ ?

As far as I understand, since the limit $\lim n+1 = \infty$, we cannot use the proposition 1.4.

So what am I missing in here ?

Answer 1

You guessed it right: for Conway, $\lim_{n\to\infty}n+1=\infty$.