For any natural number $n$, Prove that $$\displaystyle \prod^{n}_{r=1}\bigg(r+\frac{1}{n}\bigg)\leq 2(n!)$$
Trial Solution: Using $\displaystyle \frac{1}{n}\leq 1,2,3,\cdots n$
$\displaystyle \prod^{n}_{r=1}\bigg(1+\frac{2}{n}\bigg)\leq 2\cdot 4\cdot 6\cdots \cdots 2n$
$$\prod^{n}_{r=1}\bigg(1+\frac{2}{n}\bigg)\leq 2^n\cdot n!$$
Could some help me how to prove my original inequality, Thanks