Monotonicity of an iterated square-root sequence

Proved

WorkbookSource.plus_8564

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksequencessource-checked

Given the sequence $\{ a_{n}\}$ where $a_{0}= \frac{1}{3}$ and $a_{n}= \sqrt{\frac{1+a_{n-1}}{2}}$ . Show that $\{ a_{n}\}$ is monotonically increasing.

Source: InternLM Lean-Workbook, record lean_workbook_plus_8564 (Apache-2.0).

Preamble

import Mathlib

Formal statement

theorem WorkbookSource.plus_8564 (a : ℕ → ℝ) (a0 : a 0 = 1 / 3) (a_rec : ∀ n, a (n + 1) = Real.sqrt ((1 + a n) / 2)) : ∀ n, a n ≤ a (n + 1)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_8564; Apache-2.0

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