Context
Both the Fulton and Cox/Little/Schenck books on Toric Varieties include an exercise to show that if an inclusion $\tau\subset \sigma$ of cones induces an open embedding $U_\tau \hookrightarrow U_\sigma$, then $\tau$ is a face of $\sigma$.
This answer tries to answer this exercise using only the earlier material in Fulton, but I am interested in a solution that uses the torus action, what Fulton alludes to with his hint "If this is difficult now, try it after you have read about orbits in Chapter 3".
Problem
Cox's book has a more guided exercise, stated as follows:
3.2.10. In Proposition 1.3.16, we saw that if $\tau$ is a face of the strongly convex rational polyhedral cone $\sigma$ in $N_\mathbb{R}$, then $U_\tau=\mathrm{Spec}(\mathbb C [S_\tau])$ is an affine open subset of $U_\sigma = \mathrm{Spec}(\mathbb C [S_\sigma])$. In this exercise, you will prove the converse, i.e., that if $\tau \subseteq \sigma$ and the induces map of affine toric varieties $\phi: U_\tau \to U_\sigma$ is an open immersion, then $\tau \preceq \sigma$, i.e., $\tau$ is a face of $\sigma$.
(a) Let $u,u' \in N\cap \sigma$, and assume $u+u'\in \tau$. Show that $$\lim_{t\to 0} \lambda^u(t) \cdot \lim_{t\to 0}\lambda^{u'}(t) \in U_\tau.$$
(b) Show that $\lim_{t\to 0}\lambda^u(t)$ and $\lim_{t\to 0}\lambda^{u'}(t)$ are each in $U_\tau$. Hint: Use the description of points as semigroup homomorphisms.
(c) Deduce that $u,u'\in \tau$, so $\tau$ is a face of $\sigma$.
Thoughts
It is clear to me that (b) implies (c), and that (c) is sufficient. I am stuck showing/interpreting (a), and seeing how to get (b).
But, for (a), what does this multiplication mean, for points not in the torus? For $t\in \mathbb C^*$, we have $\lambda^u(t), \lambda^{u'}(t) \in T$ the torus, so we have multiplication. But in the limit, we no longer land in the torus (unless $u,u'=0$). Should I interpret these points as semigroup homomorphisms $S_\sigma \to \mathbb C$, and then the multiplication is the product of the semigroup homomorphisms? Even so, I'm having trouble with knowing when some limits converge.