Word Count : 506

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To prove that the sequence \( a_n = \frac{1}{n^2 + n + 1} \) is a Cauchy sequence, we need to show that for every \( \epsilon > 0 \), there exists an \( N \in \mathbb{N} \) such that for all \( m, n > N \), the absolute difference \( |a_n - a_m| < \epsilon \). ### Step 1: Express the difference We begin by finding \( |a_n - a_m| \), where \( a_n = \frac{1}{n^2 + n + 1} \) and _____ ___ ___ ___ ________ _______ ______ _________.
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