Intereting Posts

Consistency strength of 0-1 valued Borel measures
Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$
How to determine the direction of one point from another, given their coordinates?
What is the geometrical difference between continuity and uniform continuity?
What does “if and only if” mean in definitions?
Invertibility of $BA$
Learning path to the proof of the Weil Conjectures and étale topology
Is there a procedure to determine whether a given number is a root of unity?
Proving The Average Value of a Function with Infinite Length
To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^3+a^3}\sqrt{x^3+b^3}\sqrt{x^3+c^3}}$
Finite index subgroup with free abelianization
If $e^{itx_n}$ converges for every $t\in\mathbb R$, then does $x_n$ converge?
a question about Fourier transforms
Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$
Proof of convergence of Dirichlet's Eta Function

So, I want to prove that $$\sup_{n \in \Bbb N}{\sin (n)} = 1$$

I was thinking of proving that some set related to $\pi$ is dense in $\Bbb R$ that will then imply there is some $n \in \Bbb N$ s.t. $\sin(n)$ is as close to $1$ as desired. ($\left( \frac m n \right) \times \pi\quad \forall m,n \in \Bbb N$?)

- Inequality involving the regularized gamma function
- Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$
- Applying analysis to solve a line-of-sight problem
- Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same?
- Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number
- Does $\sum_{k=0}^{\infty}\sin\left(\frac{\pi x}{2^k}\right)$ have a simple form with interesting properties?
- Limit of a particular variety of infinite product/series
- Prove that the limit definition of the exponential function implies its infinite series definition.
- Another Proof that harmonic series diverges.
- Problem similar to folland chapter 2 problem 51.

The additive subgroups of $\mathbb R$ are either dense or lattice (that is, $x\mathbb Z$ for some $x$). Since $\pi$ is irrational, the subgroup $\mathbb Z+2\pi\mathbb Z$ is not lattice hence it is dense, in particular, for every $\varepsilon\gt0$ there exists some integers $n$ and $m$ such that $|n+2m\pi-\frac12\pi|\leqslant\varepsilon$.

If $n\geqslant0$, this yields $\sin(n)\geqslant\cos(\varepsilon)$. If $n\lt0$, note that $|-3n-6m\pi+2\pi-\frac12\pi|\leqslant3\varepsilon$ hence $\sin(-3n)\geqslant\cos(3\varepsilon)$ with $-3n\geqslant0$.

Since $\varepsilon\gt0$ is arbitrary and $\cos(\varepsilon)\to1$ and $\cos(3\varepsilon)\to1$ when $\varepsilon\to0$, this proves the result.

Assume that isn’t the case, i.e. $1 > \sup_{n \in \Bbb N}{\sin (n)} = 1 – \epsilon$ for some $\epsilon > 0$.

Let $\delta = \pi/2 – \arcsin(1 – \epsilon)$. Then $|1 – \sin(n)| < \epsilon$ if $|n-(2\pi k + \pi/2)| < \delta$ for some $k\in \mathbb N$, which is equivalent to

$$|2\pi k – n| < \delta + \pi/2$$

It now follows from Dirichlet’s approximation theorem that you can always find $n,k$ such that this condition is fulfilled. Contradiction!

*Edit: Actually, it seems that there should be an elementary justification for the last step since we don’t even need the magnitude estimate from Dirichlet’s theorem, but I can’t think of one right now.*

Another option using the equidistribution theorem, though this is something of a sledgehammer given the other answers: use the fact that $1 / (2 \pi)$ is irrational, so the sequence $n / (2 \pi)$ for $n \in \mathbb{N}$ is equidistributed modulo $1$. Therefore $n / (2 \pi) \pmod{1}$ visits points arbitrarily close to $1/4$ as $n \to \infty$, so $n \pmod{2 \pi}$ visits points arbitrarily close to $\pi/2$, so $\sin{n}$ takes values arbitrarily close to $1$ as $n \to \infty$ (by continuity and periodicity of $\sin$).

- Exercise on $\dim \ker T^i$, with $T^4=0$
- Are projections onto closed complemented subspaces of a topological vector space always continuous?
- Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$
- Should I understand a theorem's proof before using the theorem?
- Let $ A $, $ B $ and $ C $ be sets. If $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $, then show that $ B = C $.
- Inducing orientations on boundary manifolds
- If there is a unique left identity, then it is also a right identity
- Find the side of an equilateral triangle given only the distance of an arbitrary point to its vertices
- Midpoint-Convex and Continuous Implies Convex
- Any idea about N-topological spaces?
- Is statistical dependence transitive?
- Number of elements of order $p$ is a multiple of $p-1$ (finite group).
- How to derive the Golden mean by using properties of Gamma function?
- Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite
- Can the complex numbers be realized as a quotient ring?