Without using the methods of the entry determining series convergence, we show that the real-term series $$\sum_{n=0}^\infty\frac{1}{n!} \;=\; 1+\frac{1}{1!}+\frac{1}{2!}+\ldots$$ is convergent by using Cauchy criterion for convergence, being in force in $\mathbb{R}$ equipped with the usual absolute value $|.|$ as norm.

Let $\varepsilon$ be an arbitrary positive number. For any positive integer $n$ , we have $$\frac{1}{n!} \;\leqq\; \frac{1}{1\cdot2\cdot2\cdots2} \;=\; \frac{1}{2^{n-1}},$$ whence we can estimate as follows.