\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Use the mean value theorem to prove that for every \(x > 0\), \(\sqrt{x + 1} - \sqrt{x} = \dfrac{1}{2 \sqrt{c}}\) for some \(c \in ]x, x + 1[\). Deduce the inequality \(\sqrt{x + 1} - \sqrt{x} < \dfrac{1}{2 \sqrt{x}}\).
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