Given a convex absorbing set $A$ in a vector space $X$, the Minkowski functional of $A$ at $x$ is defined to be $$\mu_{A}(x) = \left\{ t > 0 : \frac{x}{t} \in A \right\}.$$

One property of this functional is that it satisfies is subadditivity, $$\mu_{A}(x + y) \leq \mu_{A}(x) + \mu_{A}(y)$$ for all $x, y \in X$.

In all the proofs I've seen for this, the final inequality is always $$\mu_{A}(x + y) \leq \mu_{A}(x) + \mu_{A}(y) + 2\varepsilon \hspace{20pt}\forall\varepsilon > 0,$$ and the statement that follows is "This implies the desired inequality". Of course, I can see why this is the case (take $\varepsilon \to 0$), but how does one show that formally? What mathematical steps prove the implication?

We want to prove $$B\leq C+2\epsilon \quad\forall \epsilon>0 \Rightarrow B\leq C,$$ where $B=\mu_{A}(x+y)$ and $C=\mu_{A}(x)+\mu_{A}(y)$.
The proof proceeds by contradiction: suppose $B>C$. Then $\epsilon_0:=\frac{B-C}{4}>0$, so it is a valid choice for $\epsilon$. Hence we obtain $$B\leq C+2\epsilon_0=\frac{B+C}{2}<\frac{B+B}{2}=B,$$ an obvious contradiction.