Almost a perfect square

The question is about the factorials . We have

$\begin{array} { l l l } 1! & = 1 & = 1 \\ 2! & = 1 \times 2 & = 2 \\ 3! & = 1 \times 2 \times 3 & = 6 \\ 4! & = 1 \times 2 \times 3 \times 4 & = 24 \\ 5! & = 1 \times 2 \times 3 \times 4 \times 5 & = 120 \\ 6! & = 1 \times 2 \times 3 \times 4 \times 5 \times 6 & = 720 \\ 7! & = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 & = 5040 \\ 8! & = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 & = 40320 \\ \end{array}$

Since the factorials are increasing, hence the only 4-digit number is 5040.