Merry fuck you eve. This one is quick

Let Y be a subspace of X, where X is Noetherian. We want to show that Y is Noetherian.

Suppose we have a chain

(1)

of closed subsets of Y

Note, by the subspace topology, for each i = 1,2,…:

(2)

for some C_i closed in X.

For each i, let

(3)

and note that each D_i is closed in X.

Then the chain

(4)

stabilizes since X is Noetherian. So for some n,

(5)

Note: ∀k < i, Y _k ⊃ Y _i, so

Y i	= Y i ∩ Y i-1 ∩∩ Y 1			(6)
	= (Y ∩ Ci) ∩ (Y ∩ Ci-1) ∩∩ (Y ∩ C1)	(By (2))		(7)
	= Y ∩ Ci ∩ Ci-1 ∩∩ C1			(8)
	= Y ∩ Di			(9)

But intersecting Y with (5) yields

Y ∩ Dn = Y ∩ Dn+1 =		(10)
Y n = Y n+1 =	(using (9))	(11)
		(12)

So we're done. WELP, THE EQUATION NUMBERS ARE ALL FORMATTED FUCKED UP, BUT FUCK IT.