# Induction proof given recurrence of algorithm

I am having trouble starting this proof and wanted some clarification. Here are the details given:

ArrayList<T> sort (ArrayList<T> A) {
if (A.size() <= THRESHOLD) {
ArrayList<T> B = copy(A)
insertionSort(B)
returnB
} else {
ArrayList<T> B1 = new ArrayList<T>()
ArrayList<T> B2 = new ArrayList<T>()
split(A, B1, B2)
ArrayList<T> C1 = sort(B1)
ArrayList<T> C2 = sort(B2)
return merge(C1, C2)
}
}


Let $$T_{THRESHOLD}(n)$$ be the maximum number of steps executed by the algorithm when executed on an input array with a positive length $$n$$. All of the constants used in the equations are all greater than 0.

$$\begin{equation} T_{THRESHOLD} (n) \leq \begin{cases} \alpha n^2 + \beta n + \delta & \text{if n \leq \mathrm{THRESHOLD}}\\ T_{THRESHOLD}(\lceil n/2 \rceil,k) + T_{THRESHOLD}(\lfloor n/2 \rfloor,k) + \gamma n + \zeta & \text{if n > \mathrm{THRESHOLD}}\\ \end{cases} \end{equation}$$

and for positive integers $$n$$ and $$k$$:

$$\begin{equation} T(n,k) = \begin{cases} \alpha n^2 + \beta n + \delta & \text{if n \leq k}\\ T(\lceil n/2 \rceil,k) + T(\lfloor n/2 \rfloor,k) + \gamma n + \zeta & \text{if n > k}\\ \end{cases} \end{equation}$$

Then it is asked to prove that if THRESHOLD is a positive integer then $$T_{\mathrm{THRESHOLD}}(n) \leq T(n, \mathrm{THRESHOLD})$$ for every positive integer $$n$$.

I was then wondering if it is required to use Constructive Induction for a question like this since the recurrence equations have been provided. Is it also necessary to solve for variables $$\alpha, \beta, \delta$$ for solving the basis and induction step of the proof? As well, how would you go about doing the induction with two variables (THRESHOLD and $$n$$). Anything helps, thank you.