Showing that $f(x)$ is increasing on $(0,+\infty)$ I am collecting some easy problems for my students and now I am facing to the following problem:

Prove that the function $$f(x)=\left(1+\frac{1}{x}\right)^x$$ is increasing in   $(0,+\infty)$.

Undoubtedly, they will solve it by using the logarithmic differentiation. I am wonder what can I do if someone wants me to verify it just by doing the definition of increasing function? I think , I am missing somethings here around. Light my way. Thanks!

You can use the Bernoulli inequality $(1+x)^\alpha \geq 1+\alpha x$ for real $x>-1$ and $\alpha \geq 1$.  Then for $x>0$ and $\alpha \geq 1$
$$
\alpha x \log(1+\frac{1}{\alpha x}) = x\log(1+\frac{1}{\alpha x})^\alpha \geq x\log(1+\frac{1}{x}).
$$
Edit:  See E.Lim's comment above.  The use of Bernoulli's inequality could be questionable.
