rule ∞

Rozaŭtuno@lemmy.blahaj.zone to 196@lemmy.blahaj.zone – 552 points –

a

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What is ∞ + ∞?

Let x_n be an infinite, real sequence with lim(n -> ∞) x_n = ∞.

Let y_n be another infinite, real sequence with lim(n -> ∞) y_n = ∞.

Let c_n be an infinite sequence, with c_n = 0 for all n ∈ ℕ.

Since y_n diverges towards infinity, there must exist an n_0 ∈ ℕ such that for all n ≥ n_0 : y_n ≥ c_n. (If it didn't exist, y_n wouldn't diverge to infinity since we could find an infinite subsequence of y_n which contains only values less than zero.)

Therefore:

lim(n -> ∞) x_n + y_n ≥ lim (n -> ∞) x_n + c_n = lim(n -> ∞) x_n + 0 = ∞

□

So the answer is □?

In case you aren't joking, '□' is used to indicate the end of a mathematical proof. It's equivalent to q.e.d

I was not joking, which also probably explains why I have no idea what anything else in your post says.

No worries, I made the comment mostly for people with somewhat advanced knowledge in math. A year ago I wouldn't have understood any of it either.

i think this means that ∞ + ∞ > ∞

Not quite. It's somewhat annoying to work with infinities, since they're not numbers. Technically speaking, ∞ + ∞ is asking the question: What is the result of adding any two infinite (real) sequences, both of which approaching infinity? My "proof" has shown: the result is greater than any one of the sequences by themselves -> therefore adding both sequences produces a new sequence, which also diverges to infinity. For example:

The series a_n = n diverges to infinity. a_1 = 1, a_2 = 2, a_1000 = 1000.

Therefore, lim(n -> a_n) = ∞

But a_n = 0.5n + 0.5n.

And lim(n -> ∞) 0.5n = ∞

So is lim(n -> ∞) a_n = 2 • lim(n -> ∞) 0.5n = 2 • ∞?

It doesn't make sense to treat this differently than ∞, does it?

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