Sum of two polyBound functions is bounded by a polyBound

The sum of two polynomial bounds $P_1(n) = c_1 (n+1)^{d_1}$ and $P_2(n) = c_2 (n+1)^{d_2}$ is bounded by a single polynomial bound $P(n) = c (n+1)^d$.

Specifically, for any leading coefficients $c_1, c_2 \in \mathbb{N}$ and degrees $d_1, d_2 \in \mathbb{N}$, there exist $c, d \in \mathbb{N}$ such that for all $n \in \mathbb{N}$:

Taking $c = c_1 + c_2$ and $d = \max(d_1, d_2)$ (or $d_1 + d_2$), each term satisfies $c_i (n+1)^{d_i} \le c_i (n+1)^d$ because $n + 1 \ge 1$. Summing the two inequalities yields $(c_1 + c_2)(n+1)^d$.

This closure property under addition is the analytic ingredient required to verify that sequential composition of polynomial-time Turing machine computations preserves polynomial running time.