# on infinities, sets and limits

Is it mathematically correct to define the following:

• $$[0, \infty] \equiv \displaystyle \lim_{n \to \infty}{[0, n]}$$
• $$[0, \infty) \equiv \displaystyle \lim_{n \to \infty}{[0, n)}$$

More generally, I'm concerned with the "boundary" between the arbitrarily large and the notion of infinity. Is there some good philosophical treatment of this sort of conundrum?

• Even if it were correct, why would you? The first one makes no sense and the second one is just $\{x: x\ge 0\}$. – John Douma Oct 17 '18 at 20:11
• You can define stuff as you best see fit. The important part is that, then, you don't expect the first one to be consistent with $\limsup\limits_{n\to\infty} A_n=\bigcap\limits_{n\in\Bbb N}\bigcup\limits_{k\ge n} A_k$ and $\liminf\limits_{n\to\infty} A_n=\bigcup\limits_{n\in\Bbb N}\bigcap\limits_{k\ge n} A_k$. – Saucy O'Path Oct 17 '18 at 20:11
• For [0,oo], look up extended reals. – William Elliot Oct 18 '18 at 2:33