The superior and inferior limit of $a_n= [1+\sin n]$

Question

I'm given the succession $$a_n= [1+\sin n]$$ and I should find the superior limit $ \limsup_{n \to \infty}a_n$ and the inferior limit $ \liminf_{n \to \infty}a_n$

$-1<\sin n<1$ and then $0< [1+\sin n]<2$

In my opinion it should be $ \limsup_{n \to \infty}a_n=2$

$ \liminf_{n \to \infty}a_n=0$

but in the book the suggested solution is 1 for the superior limit (the inferior is =0). Where am I making mistakes?

Are you sure that it is $[1+\sin n]$ rather than $\lfloor 1+\sin n\rfloor$? — José Carlos Santos, Dec 26 '19 at 19:24
@JoséCarlosSantos what is the difference between the floor function and [. I believed they were the same thing — Anne, Dec 26 '19 at 19:36
It was not clear to me that you knew that $[\cdot]$ could also mean the floor function. Then, it would have been more natural that you would have written$$0<\lfloor1+\sin n\rfloor\leqslant1.$$ — José Carlos Santos, Dec 26 '19 at 20:06
@JoséCarlosSantos ok now I see, I considered the value 2 but $\sin n<1$ so , as you wrote, $0<[1+ \sin n] \le 1$ and my book is right — Anne, Dec 26 '19 at 20:34

Christian Blatter · Answer 1 · 2019-12-27T09:26:02.443

I guess we are supposed to interpret $[t]$ as $\lfloor t\rfloor$, i.e., as the largest integer $\leq t$.

Given that all real numbers $x$ where $\sin x=1$, i.e. the numbers $\bigl(2j+{1\over2}\bigr)\pi$ with integer $j$, are irrational, it follows that all values $\sin n$ $\>(n\in{\mathbb N})$ are in fact $<1$. Therefore $\lfloor 1+\sin n\rfloor\leq1$ for natural $n$, and this implies $\limsup_{n\to\infty}\lfloor1+\sin n\rfloor\leq1$.

On the other hand $\sin x>0$ in infinitely many disjoint intervals of length $>3$. This implies that there are infinitely many $n$ with $\sin n>0$, hence $\lfloor1+\sin n\rfloor\geq1$, so that $\limsup_{n\to\infty}\lfloor1+\sin n\rfloor\geq1$.

These two paragraphs show that $\limsup_{n\to\infty}\lfloor1+\sin n\rfloor=1$.

Isn't there a more elementary reason? – Botond Dec 26 '19 at 19:54 — Botond, Dec 26 '19 at 19:54

The superior and inferior limit of $a_n= [1+\sin n]$

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