No. of Interesection of the graph of $y=\sin x$ and $y=x/n$

Question

Let $a_n$ denotes the number of times the line $y=x/n$ cuts the graph of $y=\sin x$. Then which of the following are true?

$(a)a_n$ is a Cauchy sequence.

$(b) a_n$ has a convergent subsequences.

$(c) \inf\{a_n\}$ is $1$.

$(d)$ All terms of $a_n$ are distinct.

My attempt

I am going to show $a_n$ has unbounded subsequence and so $a_n$ is unbounded.

Let $n_k=\lceil (2k+1)\pi\rceil $ where $k\in \mathbb{N}$

Then $a_{n_k}\ge 2(2k+1)+1=4k+3$ since the origin is always there and $\sin x$ is odd function.

So $(a)$ is false.

We have $a_1=1, a_2=a_3=a_4=3$

So $(c)$ is true and $(d)$ is false

Now, intuitively it's clear to me $a_n$ is non-decreasing and so if a subsequence is convergent ,it must be bounded above.

I think there is no convergent subsequence and for that I need to prove every subsequence of $\{a_n\}$ is unbounded above.Am I right?

Please help me complete the problem. Also I was thinking whether a general formula exist for $a_n$.?

Thanks for your time

Answer 1

What you did is fine. Note that the sequence $(a_n)_{n\in\Bbb N}$ is increasing (although, as you noticed, not strictly increasing). And every subsequence of an increasing unbounded sequence is also increasing and unbounded. Therefore, it cannot be convergent.