Let \(L\) denote the set of plurisubharmonic functions \(u\) on \(\mathbb C^n\) of logarithmic growth, that is \(u(z) \leq \text{log }^+|z|+C\). For a bounded Borel set \(E\) in \(\mathbb C^n\), define \(V_E(z) = \sup\{u(z): u\in L, u\leq 0 \text{ on } E\}\). If \(K\) is a compact subset of \(\mathbb C^n\), then \(V_K\) can be obtained by \[ V_K(z) = \max\left[ 0, \sup \left\{\frac 1{\text{deg }p}\log|p(z)|: p \text{ hol. polynomial }, \|p\|_K\leq 1 \right\}\right]. \] For each positive integer \(n\), let \[ V_K^{(n)}(z) = \sup\left\{ \frac 1{\text{deg }p} V_{p(K)}(p(z)): p \text{ hol. polynomial },1\leq \text{ deg }p \leq n\right\}. \] It is known that the functions \(V_K^{(n)}\) increase to the function \(V_K\). For \(u\in L\), the Robin function \(\rho_u\) of \(u\) is defined as \( \rho_u(z) = \limsup_{|\lambda|\to \infty}[u(\lambda z) - \log|\lambda|]\). The main result of the paper is that if \(K\) is regular, then all of the functions \(V_K^{(n)}\) are continuous, and that their associated Robin functions \(\rho_{V_K^{(n)}}\) increase to \(\rho_{V_K}\) for all \(z\) outside a pluripolar set. It should be noted that this last result is not an immediate consequence of the monotone convergence of the functions \(V_K^{(n)}\) to the function \(V_K\).