<s>[INST] Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing 
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.

I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$
I am trying to go ahead of this step.
 [/INST] Note that the function $f(x) = (1+\frac 1x)^x$ is differentiable.  We may find its derivative as follows:
$$
\ln(f(x)) = \ln\left((1+\frac 1x)^x\right) = x\ln(1 + \frac 1x)\\
f'(x)/f(x) = \ln\left(1 + \frac 1x\right) + \frac{x}{1+\frac 1x}\cdot \frac{-1}{x^2}\\
f'(x)/f(x) = \ln\left(1 + \frac 1x\right) - \frac{1}{x+1}\\
f'(x) = \left[\ln\left(1 + \frac 1x\right) - \frac{1}{x+1}\right]f(x)\\
f'(x) = \left[\ln\left(\frac {x+1}x\right) - \frac{1}{x+1}\right]f(x)\\
f'(x) = \left[\ln(x+1) - (\ln(x) + \frac{1}{x+1})\right]f(x)
$$
Show that the derivative is positive from $x = 2013$ to $x = 2014$.
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