Let $A\subset B\subset C$ be commutative rings. Suppose $B$ is integral over $A$, and $C$ is integral over $B$. Then I want to show that $C$ is integral over $A$.
To be integral means that for every $b\in B$, there exists a monic polynomial with coefficients in $A$ such that $b$ is a root, i.e. $b^n+a_{n-1}b^{n-1}+\cdots+a_1b+a_0=0$ for $a_{n-1},\ldots,a_0\in A$. Likewise, for any $c\in C$ we can write $c^k+b_{k-1}c^{k-1}+\cdots+b_1c+b_0=0$. Now, how can we write $c$ as a polynomial with coefficients in $A$?
This is a very standard but nontrivial result: see e.g. $\S$ 14.1 of these notes.
This occurs in the first section of a long chapter on integral extensions and is the fifth result in that section. Especially, the proof uses the fact that if $R \subset S$ is a ring extension, then $\alpha \in S$ is integral over $R$ if and only if there is a subring $T$ with $R \subset R[\alpha] \subset T \subset S$ with $T$ finitely generated as an $R$-module.