How does Lamport imply anomalous behavior is impossible given these constraints?
The constraint as written by Lamport is $C_i(t+\mu)-C_j(t)\gt0$ or $C_i(t+\mu)\gt C_j(t)$ which can be re-written  $$C_i(t+\mu) - C_i(t)\gt C_j(t) - C_i(t) \tag{*}\label{*}$$
Given Lamport's previous deductions, we recognise the boundary conditions each side must satisfy namely:
$$(1-\kappa)\mu \lt C_i(t+\mu) - C_i(t)$$
$$ \epsilon \gt C_j(t) - C_i(t)$$
Now, Lamport wants to find out the range of values for 'variables' $\epsilon,\kappa$ ($\mu$ is fixed) that will ensure the strict inequality $\eqref{*}$.
Obviously, for real numbers (Lamport's starting assumption) if $x \lt A$ and $y \gt B$, then as long as $y \leq x$ we know $A\gt B$.
In other words, as long as $\epsilon \leq (1-\kappa)\mu$ then $\eqref{*}$ is true and so $$C_i(t+\mu)-C_j(t)\gt0$$