  In this paper, we give a rigorous proof of the renormalizability of the
massive $\phi_4^4$ theory on a half-space, using the renormalization group flow
equations. We find that five counter-terms are needed to make the theory
finite, namely $\phi^2$, $\phi\partial_z\phi$, $\phi\partial_z^2\phi$,
$\phi\Delta_x\phi$ and $\phi^4$ for $(z,x)\in\mathbb{R}^+\times\mathbb{R}^3$.
The amputated correlation functions are distributions in position space. We
consider a suitable class of test functions and prove inductive bounds for the
correlation functions folded with these test functions. The bounds are uniform
in the cutoff and thus directly lead to renormalizability.
