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The first step in mathematical induction is to establish the 'base case'. This is like the foundation of a building. We must prove that the statement is true for the initial value. In our scenario, the starting point is when \( n = 4 \). We evaluate both sides of the inequality: \( 4^2 = 16 \) and \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
Since \( 16 < 24 \), the base case holds true. This step guarantees that the statement works at least for one beginning instance, which is crucial to proceed to the next steps.

The inductive hypothesis is where we make an educated assumption. Here, we assume that our statement is correct for an arbitrary integer \( k \), equal to or greater than 4. This assumption is our stepping stone. Imagine it like saying, "Let's pretend this bridge holds" before confidently crossing it.
So, we assume \( k^2 < k! \) to be true. This assumption will lead us to the next phase, where we aim to show that if the statement holds for \( k \), it must also hold for \( k+1 \).

The inductive step is where we take on the task of proving the hypothesis for \( k+1 \). This follows the logic of dominoes: if one falls, the next one will fall too. Our goal is to demonstrate that if \( k^2 < k! \) is true, then \( (k+1)^2 < (k+1)! \) must also be true. We start with the expression for \( (k+1)^2 = k^2 + 2k + 1 \) and relate it to \( (k+1)! \).
By combining the hypothesis \( k^2 < k! \) with algebra, we multiply both sides by \( (k+1) \) and rearrange to confirm \( (k+1)^2 < (k+1)! \), completing the inductive step. Checking these inequalities ensures every domino in the sequence falls, proving the statement true indefinitely for all \( n \geq 4 \).

Factorial inequalities involve comparing a factorial with another mathematical expression. In this case, the inequality we're interested in is \( n^2 < n! \). Factorials grow rapidly because each term is a multiplication of all positive integers up to \( n \). Hence, for \( n \geq 4 \), it's expected that \( n! \) quickly surpasses \( n^2 \).
Understanding this innate growth behavior of factorials helps compare them efficiently in inequalities. As \( n \) increases, \( n! \) expands faster than \( n^2 \), readily confirming the inequality for larger \( n \). This growth aspect is strategic in the proof by induction and illustrates how factorials can dominate quadratics like \( n^2 \) for sufficiently large numbers.