Fundamental theorem of linear programming

Since $P$ is a bounded polyhedron defined by finitely many linear inequalities, it has finitely many vertices $v_1,\ldots,v_k$, and a classical fact about polytopes (Minkowski's theorem) states that every point of $P$ is a convex combination of these vertices: $x=\sum_{i=1}^{k}\lambda_i v_i$, where $\lambda_i\ge 0$ and $\sum_i \lambda_i=1$.

Because the objective $c^{\mathsf{T}}x$ is linear, evaluating it on such a convex combination gives $c^{\mathsf{T}}x=\sum_i \lambda_i\left(c^{\mathsf{T}}v_i\right)\le \left(\max_i c^{\mathsf{T}}v_i\right)\sum_i\lambda_i=\max_i c^{\mathsf{T}}v_i$, since each $\lambda_i\ge 0$ and they sum to $1$.

This shows every feasible $x$ has objective value no larger than the best vertex value $\max_i c^{\mathsf{T}}v_i$. But that best vertex, call it $v_{i^*}$, is itself a feasible point of $P$, so it actually attains this upper bound: $c^{\mathsf{T}}v_{i^*}=\max_{x\in P} c^{\mathsf{T}}x$. Hence the optimum of the linear program is achieved at the vertex $v_{i^*}$, proving the claim.