Sum of factorials equation

Question

Could you explain why constant $c \gt 0$ can't satisfy equation $c + \frac{c \cdot n!}{\varphi } \le 1$ , where $\varphi = \sum_{i=0}^{n-1}i!$, where $n \to \infty$

Answer 1

Hint: $$\frac{n!}{0!+\ldots+(n-2)!+(n-1)!} \ge \frac{n!}{(n-2)!+\ldots+(n-2)!+(n-1)!} = \frac{n!}{(n-1)!+(n-1)!} = \frac{n}{2}.$$