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Ayaan Barr

Answered question

2022-07-04

How to prove

\underset{n \to \infty}{lim sup} | \sin (n) | = 1

?

Answer & Explanation

poquetahr

Beginner2022-07-05Added 18 answers

See this article, a special case is that there is two increasing sequences of odd positive integers

(p_{n}), (q_{n})

such that

| \frac{π}{2} - \frac{p_{n}}{q_{n}} | \leq \frac{1}{q_{n}^{2}}, n > 1.

Note that

\sin | x | = | \sin x |

for

x \in [0, π]

, then

\sin | \frac{q_{n} π}{2} - p_{n} | = | \cos p_{n} | < \frac{1}{q_{n}} \to 0.

therefore

| \sin p_{n} | \to 1

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