I came across the following problems during the course of my studying of real analysis:
Show that the sequence $(a_n)$ defined by $a_n = \left(1+ \frac{1}{n} \right)^{n}$ is bounded above by $3$.
I think we can use the binomial theorem. So $$a_n = \left(1+ \frac{1}{n} \right)^{n} = \sum_{k=0}^{n} \binom{n}{k} \left(\frac{1}{n} \right)^{k}$$
$$= 1+ \sum_{k=1}^{n} \binom{n}{k} \left(\frac{1}{n} \right)^{k}$$
From here, how would I deduce that this is $\leq 3$?