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 \[\left[1+\left(\frac{1}{\sqrt{n}}\right)\right]^{n-2} \geq \tfrac{1}{2}n.\]