Show $\left(1+\frac 1 x\right)^{x} < e < \left(1+\frac 1 {x}\right)^{x+1}$

I had to resort to the following characterization of number $ e$ in order to Proof $ \sum\limits_{n \le k/2} \frac 1 n < \log k$ to show Pólya's inequality:

$ \forall x > 0. \left(1+\frac 1 x\right)^{x} < e < \left(1+\frac 1 {x}\right)^{x+1}$

Unfortunately, I don’t have this result yet in my theorem prover. What is an easy way to derive it?