How to show that the convex hull of a set describes a polyhedron?

Question

I read in the solution manual of convex optimization (by Stephen Boyd) that $$C=conv\{x_1,\cdots x_k\}$$ describes the polyhedron. Where $$conv\{x_1\cdots x_k\}=\{y| y=\theta_1x_1+\cdots \theta_kx_k, \theta_i\geq0, \sum_{i=1}^{k}\theta_i=1\}$$ I know the construction of $conv\{x_1\cdots x_k\}$ involves linear equalities. But the number of inequalities are infinite due to the infinitely many possibilities in which we can have $\theta_i\geq0, \sum_{i=1}^{k}\theta_i=1$. So how does $C$ describes a poyhedron?