Proving the existence of a supremum $A=\{(1+(1/n))^n \mid n\text{ is taken from positive integers}\}$
How can I prove that the set above has a supremum? I've started with an assumption that 
$(1+(1/n))^n < 3$ for every positive integer n
but I could not find a way to prove it too. Could you please give me a hint?

Outline: Using the Binomial Theorem, we find that
$$\left(1+\frac{1}{n}\right)^n \le 1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}.$$
The term $\frac{1}{3!}$ is less than $\frac{1}{2^2}$. Continue. 
