2. Let an be the sequence defined inductively by a₁ = 2 and an+1 = 2 2 an + an (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥ 2 for all n € N. (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L> 0 and L² = 2.

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