This is a problem from Ideals, Varieties, and Algorithms by Cox et. al.
Let $V=V_1\cup \cdots \cup V_r$ be a decomposition of variety into its irreducible components.
Let $\Sigma$ be the singular locus of $V$ and let $\Sigma_i$ be the singular locus of $V_i$. If each $\Sigma_i$ is a proper subset of $V_i$, then show that $\Sigma$ contains no irreducible components of $V$.
In the previous part of this problem, I showed that if a point $p$ lies in a unique irreducible component, then it is nonsingular on $V$ if and only if it is nonsingular on $V_i$.
In another previous part of the problem, I showed $$\Sigma = (\bigcup_{i\ne j} (V_i\cap V_j))\cup (\bigcup_i \Sigma_i)$$
For the current part, here is my attempt:
Suppose $\Sigma$ contains a component $V_i$. Let $p$ be a point in $V_i$ that is not in $\Sigma_i$. Then $p$ is nonsingular in $V_i$ but singular in $\Sigma$. So it must lie on the intersection of two components $V_i\cap V_j$. Now I have no idea how to continue.
Thanks for any help!