Word Count : 503

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To prove the inequality: \[ 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{2(n-1)}, \] for \( n \in \mathbb{N} \) and \( n > 1 \), we will proceed by mathematical induction. ### Base Case (\( n = 2 \)): For \( n = 2 \), the left-hand side (LHS) of the inequality is: \[ 1 + \frac{1}{\sqrt{2}}. \] The right-hand side (RHS) is: \[ \sqrt{2(2-1)} = \sqrt{2}. \] We need to show that: \[ 1 + \frac{1}{\sqrt{2}} \geq \sqrt{2}. \] Calculating the numerical values: \[ 1 + \frac{1}{\sqrt{2}} \approx 1 + 0.7071 = 1.7071, \] and \[ \sqrt{2} ________ ____ ______ __________ ___ _________.
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