# Proving $\lim_{n \to \infty} \frac{\sin n^2}{\sqrt n} = 0$

I ran into a block when trying to show that $\lim_{n \to \infty} \frac{\sin n^2}{\sqrt n} = 0$. Using the definition, I wrote,

$$\left| \frac{\sin n^2}{n^{1/3}} - 0 \right| < \varepsilon \quad \text{for } \varepsilon > 0$$

subtract zero and notice $n^{1/3} > 0 \quad \forall n \in \mathbb{N}$,

\begin{align*} \frac{ \left| \sin n^2 \right| }{n^{1/3}} &< \varepsilon \\ \left| \sin n^2 \right| &< n^{1/3}\varepsilon \\ \left| \sin n^2 \right|^3 &< n\varepsilon^3 \end{align*}

I know we need to be able to find $n$ given $\varepsilon$. I'm not sure how to isolate $n$ or proceed from here.

• use that $$|\sin(n^2)|\le 1$$ Jun 15, 2017 at 17:37
• $n$ is going to $\infty?$ Jun 15, 2017 at 17:37
• @nouret yes indeed Jun 15, 2017 at 17:38

Note that $|\sin n^2|\leq 1$ for any $n\in \mathbb{N}$. Now $|\sin n^2 /\sqrtn|\leq 1/\sqrtn\leq 1/m_n$, where $m_n$ is largest integral value less than or equals to $\sqrtn$. Now you can always find a $m_n$ for a given value of $\varepsilon >0$ such that $1/m_n\leq \varepsilon$(Why?) which leads to $\lim_{n\to\infty}\frac{\sin n^2}{ \sqrtn}=0$.
Suppose does not hold then for all $N\in \mathbb{N}$, $\varepsilon <1/N\Rightarrow N<1/\varepsilon$. But for a fixed $\varepsilon$, $N_{\varepsilon}+1\geq 1/\varepsilon$, and $N_{\varepsilon}$ always exists,(where $N_{\varepsilon}$ is largest integral value less than or equals to $1/{\varepsilon}$), which contradicts our assumption. $\blacksquare$