Intereting Posts

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis
Is a Bijection From a Group to Itself Automatically an Isomorphism If It Maps the Identity to Itself?
Prove that limit $\lim_{n\to\infty}\sqrt{4+\frac{1}{n^2}}+\sqrt{4+\frac{2}{n^2}}+\cdots+ \sqrt{4+\frac{n}{n^2}}-2n=\frac{1}{8}$
Find the expected number of two consecutive 1's in a random binary string
Division polynomials of elliptic curves
Does $\int_{1}^{\infty} g(x)\ dx$ imply $\lim\limits_{x\to \infty}g(x)=0$?
How many different permutations are there of the sequence of letters in “MISSISSIPPI”?
How to prove $\frac 1{2+a}+\frac 1{2+b}+\frac 1{2+c}\le 1$?
How to solve the complex ODE $\mu f'(x) = if(x)$ in the interval $$?
Series expansion for $e^{- x^2}$
advanced calculus, differentiable and limit problem
Surface Element in Spherical Coordinates
Various proofs of Hardy's inequality
How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$?
Obtaining irrational probabilities from fair coins?

We were given a challenge by our calculus professor and I’ve been stuck on it for a while now.

Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).

What would be a good way to start?

- How to make a smart guess for this ODE
- How do “Dummy Variables” work?
- Show $\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$
- Integral involving the hyperbolic tangent
- Evaluate $\lim_{x \rightarrow 0} x^x$
- How to evaluate $\lim_{n\to\infty}\sqrt{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$

- Monotonic, surjective function
- Is this integral right?
- List of interesting integrals for early calculus students
- What are the practical applications of the Taylor Series?
- How to prove that $f_n(x)=\frac{nx}{1+n\sin(x)}$ does not converge uniformly on $$?
- Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$
- Limit of integration can't be the same as variable of integration?
- If $f$ is continuous at $a$, is it continuous in some open interval around $a$?
- A limit about $a_1=1,a_{n+1}=a_n+$
- How to prove $\frac{1}{2}f''(\xi) = \frac{f(a)}{(a-b)(a-c)}+\frac{f(b)}{(b-a)(b-c)}+\frac{f(c)}{(c-a)(c-b)}$

Following Qiaochu hint, I’ll try to elaborate a bit.

**Note:** This is a complete rewrite of the proof to fix a flaw pointed out by Qiaochu and make it overall clearer.

First some notation. Let $S^1 = [0, 2\pi)$ with 0 and $2\pi$ identified. For two points $a, b \in S^1$ we’ll denote by $a \oplus b = a + b \mod 2\pi$ (and likewise for $\ominus$) and let $A_n = \{n \mod 2\pi\}_{n=1}^{\infty}$. Also denote by $U_{\varepsilon}(p)$ a punctured $\varepsilon$-neighborhood of $p \in S^1$.

Our strategy will be to show that $\{A_n\}$ has at least one limit point in $S^1$. From this we will conclude that 0 is also a limit point and finally we will use this fact to show that every point in $S_1$ is a limit point of $A_n$ (this means that $\{A_n\}$ is dense in S^1). Having established this, we will use continuity of $\sin$ to finally resolve the problem.

**Proof**

As a preparation we will note the relation $A_{n+m} = A_n \oplus A_m \quad (1)$. As a corollary of this we have that the sequence $\{A_n\}$ is injective (for if not, there would exist $n, m \in \mathbb{N}, k \in \mathbb{Z} \quad n>m$ such that $A_n = A_m$ and so $n – m = 2 k \pi$, a contradiction with irrationality of $\pi$). This implies simple but crucial fact that the image of the sequence contains infinitely many points.

Now, to establish the density of ${A_n}$ in $S^1$ we will first show that there exists at least one limit point $p \in S^1$. This is established by a standard argument: consider intervals $I = [0, \pi]$ and $J = [\pi, 2\pi]$ and take the one which contains infinitely many points of $\{A_n\}$ (if both do, take the “bottom” one, i.e. $I$). Call this interval $I_0$. Now again divide it in two intervals of same length and let $I_1$ be the one with infinitely many points. Continue in this way to obtain an infinite sequance of intervals $I_0 \supset I_1 \supset \cdots$. Then the set $K = \cap_{n=0}^{\infty} I_n$ is non-empty (this should be covered in standard calculus course, I hope) and any point $p \in K$ is surely a limit point (by construction of $\{I_n\}$).

Thanks to the above we now know that for each $\varepsilon > 0$ there exist infinitely many points in $U_\varepsilon(p)$, i.e. infinitely many $n, m \quad n > m$ such that $|A_n – A_m – 2k\pi| < 2\varepsilon$ for some $k \in \mathbb{Z}$. But from (1) we get that $|A_{n-m} – 2k\pi| < 2\varepsilon$. Because we identify 0 and $2\pi$ in $S^1$ we can see that these differences are concentrated around 0. Therefore we have shown that 0 is also a limit point.

Now we will show that every $p \in S^1$ is a limit point. Suppose we are given $\varepsilon > 0$. We just take any $A_k$ from $U_{\varepsilon}(0)$ and note that for $l = \lfloor {p \over A_k} \rfloor$ we have $A_{lk} \in U_{\varepsilon}(p)$.

To conclude the answer we will show that any $r \in [-1, 1]$ is indeed a limit point of $\{\sin(n)\}$. Take $s = \arcsin (r) \mod 2\pi$. Then by the above, we know that $s$ is a limit point of $\{A_n\}$ so there is a subsequence $\{A_{n_k}\} \to s$ and observe by continuity of $\sin$ that $\{\sin(A_{n_k})\} = \{\sin(n_k)\} \to sin(s) = r$.

Suppose you want to find positive integers $n, k$ such that $|n – 2 \pi k – \arcsin r| < \frac{1}{m}$ for some large positive integer $m$ and $r \in [-1, 1]$. Then I claim that some value of $n$ less than $2\pi m$ works. This is the pattern I was trying to get you to see, although I wasn’t doing a very good job of it (I should’ve asked you to plot $k$ instead of $n$).

Unfortunately I think this is a little hard to prove except when $r = 0$; fortunately, using the second half of Marek’s argument you can prove the same result with a weaker bound on $n$ for general $r$ using the case of $r = 0$, so I would encourage you to try that case first. To do this, try plotting the fractional parts of the numbers $\frac{n}{2\pi}$ in $[0, 1)$.

- Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB
- Closed form for ${\large\int}_0^1\frac{\ln^3x}{\sqrt{x^2-x+1}}dx$
- Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X_1-a_1,\ldots,X_n-a_n)$?
- Why can a matrix whose kth power is I be diagonalized?
- Prove that $\frac{x^x}{x+y}+\frac{y^y}{y+z}+\frac{z^z}{z+x} \geqslant \frac32$
- How to show that the set of points of continuity is a $G_{\delta}$
- Finding a combinatorial proof of this identity: $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$
- Prove that $f$ is one-to-one iff $f \circ h = f \circ k$ implies $h = k$.
- Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.
- How to solve the irrational inequality?
- Solving complex numbers equation $z^3 = \overline{z} $
- Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A – B $
- If a hyperbola whose foci are (–2, 4) and (4, 6) touches y–axis then equation of hyperbola is
- Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
- $\epsilon$ – $\delta$ definition of a limit – smaller $\epsilon$ implies smaller $\delta$?