Htykuut

I was not allowed to edit the previous question because people spent their time and energy constructing an answer, therefore I have asked it again, with my initial error in formulating my answers corrected.

Is it true that this example I came up with, namely the sequence $$a_n=cosleft(frac{pi}{3} + frac{pi n}{2}right)$$
Has exactly $4$ limit points/accumulation points? What I tried to do was symmetrically pick an infinite amount of points that lie on the unit circle and all hop between four exact points $pm frac{1}{2}$ and $pm frac{1}{2} sqrt{3}$.

Is equality fine for accumulation points is basically what I am asking.

I also was thinking of an example with infinitely many accumulation points and wondered if the same example would cut it:

$$b_n = n cos(n) $$
Since it oscillates back and forth it comes back to every single point eventually.

As an alternative way of getting four accumulation points I could define unions of the sets $s_a= { a+ frac{1}{n}| n in mathbb{N} }$ for values $a=0, 1,2,3$ but this example feels very contrived and I do not know how to formulate this in terms of a sequence other than just listing the elements one by one in set notation: ${ frac{1}{1}, 1+frac{1}{1}, 2+frac{1}{1} dots, 1+ frac{1}{n}, 2+ frac{1}{n} dots }$

We have that

• $n=1 implies a_1=cosleft(frac{pi}{3} + frac{pi }{2}right)=-frac{sqrt 3}2$


• $n=2 implies a_2=cosleft(frac{pi}{3} + piright)=-frac12$


• $n=3 implies a_3=cosleft(frac{pi}{3} + frac{3pi }{2}right)=frac{sqrt 3}2$


• $n=4 implies a_4=cosleft(frac{pi}{3} + 2piright)=frac12$


• $n=5 implies a_5=a_1$

For the second question, yes of curse $b_n$ oscillates diverging assuming values $in mathbb{R}$ and $cos n$ is dense in $[-1,1]$. From that to conlcude that $ncos n$ is dense on the real line is not so trivial, see the related

• Is $n sin n$ dense on the real line?


• Does this sequence $a(n) = frac{1}{n^3sin(n)}$ converge


• Is it true that $varliminf_{n rightarrow +infty} |n sin n| = 0$
  


• Convergence of $sum(n^3sin^2n)^{-1}$