I see this everywhere, but cannot find a proof of it. Is it just this easy; it feels wrong.
Suppose $X$ is noet and $Y\subset X$ with subspace topology. If $Y$ has a descending chain of closed sets $Y_1 \supset Y_2 \supset Y_3 \supset \ldots$, where each $Y_i$ is formed by intersection some closed set in $X$ with $Y$. That is $(X_1\cap Y)\supset (X_2\cap Y) \supset \ldots$ but I can't seem to finish this because it is not as if I can just remove the intersections with $Y$ right?
On
This is (Bourbaki, Alg. comm., chap. II, §4, n° 2, prop. 8). Take a decreasing chain $(Y_n)$ of subsets of $Y$, closed in $Y$; note $Y_n = \overline{Y_n} \cap Y$, and the closures $\overline{Y_n}$ form a decreasing sequence of closed subsets of $X$. As this sequence stabilizes, so does $(Y_n)$.
If $Y_1\supseteq Y_2\supseteq \cdots$ is a descending sequence of closed sets in $Y$, let $Y_i=X_i\cap Y$, where $X_i$ is closed in $X$. Then the sequence $X'_i:=X_1\cap\cdots \cap X_i$ gives the same intersections with $Y$ as $X_i$ does and is a descending sequence of closed sets in $X$. Therefore, $X'_i$ must become eventually constant, so $Y_i$ must as well.