How can it be that ∞ always has an ) in interval notation?

Because you can’t include everything. It never ends, how can it be fully contained?

Well. The parenthesis is used to denote that you approach the number, as opposed to the bracket denoting that you reach that number.

No matter what number you have, no matter how high it is, you will never reach “infinity”. You can only move towards it. You can reach -3. You can reach 5,000. You can reach 7,849,503,845,298. You can’t reach infinity.
After saying “reach” so much, it doesn’t even look like a word any more.

@Sarcasm OOOOOOOOOooooooooooooohhhhhhhhhhhhhhh. Ok. Now I get it. So much better. GA.

Also note that ∞ is not itself a member of the set of real numbers.

@hiphiphopflipflapflop Is it an imaginary number, or something all to itself?

Remember to stick to defintions. You live and die by them in mathematics. Recall that imaginary numbers have their own definition. They are the set of all real multiplies of i, the sqrt(-1). So, no, ∞ is not a member of that set either.

The appearance of ∞ in interval notation is a shorthand way of saying include all reals above the real number beforehand (a, ∞) or all reals below the one afterward (-∞, b).

The definition of a closed interval is that it contains its endpoint. As these intervals have no endpoint to contain, they are regarded as open (on the relevant side)(the example you gave [3, ∞) is a closed on left-side and open on the right-side).

@hiphiphopflipflapflop We haven’t gotten to imaginary yet. Because we’re protesting Evil (j/k).

Well, I’m partly wrong. Such intervals are not considered ‘open’ but instead ‘unbounded’ on the +∞ or -∞ side.