I know that $a_n=1+\frac1n$ converges to $1$.
How do you prove that $b_n=\left(1+\frac{1}{n}\right)^c$ converges to $1$ where $c\in \mathbb{N}$ is a constant(unfortunaley originally I wrote $c\in\mathbb{R}$ which led to all the comments), and why doesn't the same proof work for $c_n=\left(1+\frac{1}{n}\right)^n$ which converge to $e$.
Please try to prove it using elementary tools (such as the definition of limit, limit arithmetic ...).
Thank you very much.