Can we find an inequality from the expansion of $[ 1+1/\sqrt{n}]^n$?

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By considering the first three terms of the binomial expansion of \(\left[ 1+\left(\dfrac{1}{\sqrt{n}}\right) \right]^n\), where \(n\) is an integer greater than \(1\), prove that \[\left[1+\left(\frac{1}{\sqrt{n}}\right)\right]^{n-2} \geq \tfrac{1}{2}n.\]