It was shown in here that $\left(1+\frac{1}{n}\right)^n < n$ for $n>3$. I think we can be come up with a better bound, as follows:
$$\left(1+\frac{1}{n}\right)^n \le 3-\frac{1}{n}$$ for all natural number $n$.
The result is true for all real number $\ge 1$, which can be shown using calculus. I wonder if the above result can be proved using mathematical induction?
I have tried but fail! Anyway, this question is also inspired by, and related to this question.
Edit:
I also found that $$\left(1+\frac{1}{n+k}\right)^n \le 3-\frac{k+1}{n}$$ for all natural number $k$, some large $N$ and $n > N$. This implies that $$\left(1+\frac{1}{2n}\right)^n \le 2-\frac{1}{n}.$$
And again, I can't prove any of them using Mathematical Induction.