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Winter boots that look amazing AND keep toes toasty? Click to find them! - Databee Business Systems

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 b_1 = 1, \quad b_{n+1} = f(b_n), \quad \text{where } f(u) = u - \frac{u^4}{4}.

📌 We aim to find $ \lim_{n \to \infty} b_n $.

📌 First, observe that $ f(u) < u $ for all $ u > 0 $, since $ \frac{u^4}{4} > 0 $. Thus, the sequence $ b_n $ is strictly decreasing and bounded below by 0. Therefore, it converges to some limit $ L \geq 0 $.

📌 Taking limits on both sides of the recurrence:

📌 \lim_{n \to \infty} b_{n+1} = \lim_{n \to \infty} f(b_n) = f(L).

📌 But since $ \lim_{n \to \infty} b_{n+1} = \lim_{n \to \infty} b_n = L $, we must have

📌 L = f(L) = L - \frac{L^4}{4} \Rightarrow \frac{L^4}{4} = 0 \Rightarrow L = 0.

📌 Thus, the sequence converges to $ \boxed{0} $.

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