An inequality $\,\, (1+1/n)^n<3-1/n \,$using mathematical induction

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.

Solution 1:

I prove this inequality does not use induction, but I think this proof also elementary proof, because this proof does not use calculus.

$$\begin{array}{lcl} \left( 1+\frac{1}{n}\right)^n &=& 1+\binom{n}{1} \frac{1}{n}+ \sum_{k=2}^n\binom{n}{k} \frac{1}{n^k}\\ &=&2+\sum_{k=2}^n \frac{1}{k!} \frac{n(n-1)\cdots (n-k+1)}{n^k}\\ &\le& 2+\sum_{k=2}^n \frac{1}{k!} \le 2+\sum_{k=2}^n \frac{1}{k(k-1)} \\ &=& 3-\frac{1}{n} \end{array} $$