Word Count : 387

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To show that the sequence \(\left( \frac{1}{n^{2}+n+1} \right)_{n \in \mathbb{N}}\) is a Cauchy sequence, we need to show that for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for all \(m, n \geq N\), the difference between the terms of the sequence is less than \(\epsilon\). That is, we need to show: \[ \left| \frac{1}{n^2 + n + 1} - \frac{1}{m^2 + m + 1} \right| < \epsilon \quad \text{for all} \, m, ___ _______ ________ _________ ___ _______ __________ _________ ________ _______ ______ ___.
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