Inductive Proof of Convergence in a Recursive Sequence
Let \(a_1 = 1\) and for all \(n \geq 1\) define \(a_{n+1} = \frac{1}{2}a_n + 1\). Prove by induction that \(a_n < 2\) for all \(n \in \mathbb{N}\).
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