I want to prove the following lemma:
Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety.
My idea is as follows, we can write $$ X = \mathcal{V}(f_1,...,f_m)$$ as the set of zeroes of homogeneous polynomials $f_1,...,f_m$. Since $W$ is Zariski closed and $W \subset X$ we can write $$ W = \mathcal{V}(f_1,...,f_m, g_1,...,g_l)$$ where $g_1,...,g_l$ are homogeneous polynomial. I need to show it is a projective variety using the fact that it is irreducible.
However, from the book [Hulek, Elementary Algebraic Geometry] Definition 2.6 $W$ is a projective variety (subset of $\mathbb{P}^n$ which is a set of zeroes of set of homogeneous polynomial equations). Where does the role of irreducibility take place? I didn't use it, at least explicitly. Is my proof correct?