Phew, an easy peasy exercise (or part of an exercise). Let's finish this baby off.

Let X be Noetherian, and suppose we're given an open cover

(1)

First, pick any V ₁ = X_i from the cover.

And now, more generally, suppose we have picked V ₁,…,V _n from the cover. Let

(2)

Then pick any X_i ∈ S_n and let V _n+1 = X_i

Now, if S_n was always nonempty, we can form a chain

(3)

(by the axiom of dependent choice, which I freaked out about in the last post) which would contradict Noetheriannness of X (see 1.7a). Hence S_n = ∅ for some n.

i.e.

∀i ∈ I: V 1 ∪∪ V n	= V 1 ∪∪ V n ∪ Xi
V 1 ∪∪ V n	= ⋃ i∈IV 1 ∪∪ V n ∪ Xi
	= V 1 ∪∪ V n ∪⋃ i∈I Xi
	= V 1 ∪∪ V n ∪ X
	= X

So X = V ₁ ∪∪ V _n is a finite subcover. Done.

Alright, so let's say you're a shut-in and you want to get outside more. At the very least: You want to get to the point that you're walking for 1 hour per day. Well, obviously, you're not going to be able to change overnight, right? You want to get to that goal via a slow, steady progression. So after a week you go from walking 0 minutes a day to 5 minutes a day. Small improvement: That's good! And furthermore, 1 week later, you go from walking 5 minutes a day to 10 minutes a day. Wow, good job, you made it to 10 minutes a day, and you maintain that rate for a whole week. You're on track! But, ah, you then hit one of those "depressive episodes". And, as a result, your 10 minutes a day recedes into 5 minutes a day for the next week. No matter, no matter. 5 minutes is better than 0 minutes, at least. Let's see if we can at least maintain that. So, for the next 2 weeks, you stay stuck in that depressive rut. Hey, it's depression, who can blame you? At least you're maintaining your 5 minutes per day. And finally, at the end of those 2 weeks, you crawl out of that "depressive episode". Well, you're still depressed overall and suffer through crippling mental problems, but at least you're keeping it at a manageable level. Manageable enough to increase your rate back to 10 minutes per day. Fantastic. You suffered a setback, but you made it back to 10 minutes. A pat on the back for you! Now, you don't want to increase your rate to 15 minutes too quickly. I mean, you just recovered from a "depressive episode" and you're still not feeling completely well. So let's maintain this 10 minutes for at least 2 weeks. Well, with a few "off-days" and "undershot days" you more or less manage those 10 minutes per day, and you increase your rate to 15 minutes per day. Holy crap! Keep up the good work. Let's see if we can maintain that. Yes, 1 week, 2 weeks, 3 weeks. Amazing! Let's bump it up to 20 minutes. 1 week, 2 weeks. By God... *Wipes tear from eye* You've gone from being a total shutin to walking 20 minutes per day. You're a third of the way to your hour per day goal. Oh, what's that? Oh, something unfortunate happened in your life? Oh... oh, no. That's triggered another... "depressive episode". Oh no, I'm so sorry! And thus due to the sapped energy from your "depressive episode", you sink back to 10 minutes a day. Your 20 minutes didn't just recede by 5 minutes, it halved. Well, that's what depression does to ya, it's an unpredictable monster, that bastard! Ah, but you maintain to manage the 10 minutes per day for a week? Great! Oh... it dipped down to 5 minutes for the next week. Well... depression is a bitch, isn't it? Better than 0 minut--oh, you just went 2 weeks with 0 minutes.... Fuck. Okay, okay. What's that? You're feeling better now? Not feeling perfect anymore, but you have a little hope? *Pats you on the back*. Alright, good luck. My advice? Increase your rate little by little. After all, you're not going to be able to change overnight, right? You have to get there via a slow, steady progression right? Start with just 5 minutes per day, and... fuck, it's been like 15 weeks already hasn't it.

Basically, this slowass saunter through section 1 isn't a good sign. HAHA.