Let $X$ be a noetherian space. We say a subset $Z$ of $X$ is constructible in $X$, if it is a finite union of locally closed subsets of $X$.
There is the following theorem of Chevalley(we are not supposed to prove it in this thread).
Theorem of Chevalley Let $X$ be a scheme. Let $Y$ be a noetherian scheme. Let $f\colon X \rightarrow Y$ be a morphism of finite type. Then $f(Z)$ is constructible in $Y$ for every constructible subset $Z$ of $X$.
Hartshorne Exercise II. 3.19 (a) is as follows. Show that the above thorem can be reduced to the following proposition.
Let $X, Y$ be affine and integral noetherian schemes. Let $f\colon X \rightarrow Y$ be a dominant morphism of finite type. Then $f(X)$ is constructible in $Y$.
How do we show this?