We analyse 4-dimensional massive $\varphi^4$ theory at finite temperature $T$ in the imaginary-time formalism. We present a rigorous proof that this quantum field theory is renormalizable, to all orders of the loop expansion. Our main point is to show that the counterterms can be chosen temperature independent, so that the temperature flow of the relevant parameters as a function of $T$ can be followed. Our result confirms the experience from explicit calculations to the leading orders. The proof is based on flow equations, i.e. on the (perturbative) Wilson renormalization group. In fact we will show that the difference between the theories at $T>0$ and at $T=0$ contains no relevant terms. Contrary to BPHZ type formalisms our approach permits to lay hand on renormalization conditions and counterterms at the same time, since both appear as boundary terms of the renormalization group flow. This is crucial for the proof.