another proof of Bernoulli’s inequality

For fixed $x>0,\\ x\eq 1$ the function

w_{x}(r)=\\int_{1}^{x}t^{r-1}dt=\\begin{cases}\\frac{x^{r}-1}{r}&r\eq 0\\\\\\log x&r=0\\end{cases}

is strictly increasing.
Bernoulli inequality is equivalent to

w_{1+x}(r)>(<)\\ w_{1+x}(1)

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