Suppose that a torus $T$ acts on a non-compact symplectic manifold $M$. Assume that this action is Hamiltonian and that the fixed point set of $T$ is compact. Let $\mu:M\to\mathfrak{t}^{*}$ denote the moment map of the action, where $\mathfrak{t}$ denotes the Lie algebra of $T$.

If there exists an $X\in\mathfrak{t}$ such that $\mu(X):M\to\mathbb{R}$ is a proper function that is bounded below, why is $\mu:M\to\mathfrak{t}^{*}$ necessarily proper?