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[Dozebôm Lolumzalìs]

Does the domain mean everything that COULD be plugged into the function? If I want to say "this makes no real, physical sense to include this in the input", what would I do?

[Dozebôm Lolumzalìs]

What are randU, randN, and randR?

[Helgoland]

Fuzzy topology indeed is a thing. There's a whole book series called "Advances in Fuzzy Systems - Applications and Theory".

[Helgoland]

Well, you need a way to connect two things A and B to get a new thing C, and a way to derive a degree of truth of C from the degrees of truth of A and B, no? Just fix a rule for that then. For example: If the truth value of A is a, and the truth value of A => C is b, assign to C the truth value a*b. You'd need some way to resolve conflicts though - maybe via taking a supremum?

[TheBiggerFish]

Oh boy.

Can I just say I have no idea what's going on anymore there.

[hops]

I'm not sure how to prove b. As I can see I don't really have x0 defined but I'm supposed to prove the claim for all n ∈ N...

[Dozebôm Lolumzalìs]

n must be >= 2, right?

[hops]

No, the given assertion that had been concluded only works for n >= 2, but I need to prove for all n ∈ N.

It's moot anyways because I'm a doof and I just realized now that I can just defined x0 by rearranging the recursive definition.

[Dozebôm Lolumzalìs]

Well, it doesn't explicitly define nx for x<0, but I guess that works.

[hops]

n can't go lower than 0 anyways because it's a natural number...

[hops]

Nevermind, I still can't solve this. I have no idea how to do this. Trying to prove it just gives me shitloads of fourth order polynomials and I mean I can probably solve it but there's probably an easy way to go about it...

EDIT: Nevermind, I overthought this.

[Arx]

Prove the base case, n=1. Then take an inductive step, which will give you an expression of something like (I don't know what notation you use) |x_k+1+1 - x_k+1|. You can use the previously accepted proposition to work from there. Does that help?

Edit: all good, then. You figured yourself out in the time it took me to find a pen and scratch out a proof.

[hops]

Well, that's what I did.

I got stuck because I was trying to prove that an absolute value of an expression is higher or equal to zero before I realized that's literally part of the definition.

[hops]

This might sound stupid, but how do I formally prove that a sequence is unbounded? I know I can easily eyeball sequences and tell if they're bounded or not, but I have no idea how to prove it.

EDIT: Nevermind, I guess I can always just prove that the absolute of the sequence's limit isn't a real number, so there can't be a real number bigger than it.

[MagmaMcFry]

More technically, you need to prove that for any real number you choose, there's a sequence element with an even larger absolute value. With this, you've shown that your sequence does not completely lie in any real interval, hence it is not bounded.