Is $\infty$ undefined? [duplicate]

Infinity does not exist in the typical real number line: it is a construct that is contained in the extended real number line, which is used in certain fields of mathematics, especially limit calculus. However, the extended real number line does not share all of the same properties of the normal real number line, which means that the apparent contradiction of 1=2 that you have produced does not hold, as certain operations which you performed to reach that point are not valid in the extended real number line. Whenever we manipulate infinity, we must enter the extended real number line, with different properties.

You may calculate with $\infty$, in a certain sense. For that purpose let $\infty$ be an object which is not a real number, e.g., $\infty:=\{\mathbb R_+\}$. (You may define $-\infty$ as $\{\mathbb R_-\}$.) Declare for any real number $r$ the set $N_r(\infty\}:=\{x\in\mathbb R\mid x>r\}$ as a neighbourhood of $\infty$.

Apply the usual definition of a limit to define if as sequence has $\infty$ as limit: in each neighbourhood of $\infty$ there are almost all elements of the sequence.

Now prove that for the limit of sequences some of the usual rules are valid, for example $\infty\cdot\infty=\infty$, which means the product of two sequences which limit is $\infty$ has limit $\infty$ as well. Also $\infty+\infty=\infty$ as well as $\infty\pm a=\infty$ for any sequence with limit $a\in\mathbb R$ and $|a|\cdot\infty=\infty$ for $a\neq0$. $a/\infty=0$ is another one for non-negative $a$.

On the other hand there are so called "indetermined expressions" as $0\cdot\infty$ or $\infty-\infty$, which can be any number (including $\infty$).