Real sequences and ceil function

Question

Let $h_n$ be a positive real sequence such that $h_n\to 0$ as $n\to\infty$.

How can I argue that

$$\left \lceil{1/h_n}\right \rceil \leq C (1/h_n)$$

for some positive constant $C$ and for all sufficiently large $n$?

My intuition says that the limit $\lim_{n\to\infty}\frac{\left \lceil{1/h_n}\right \rceil}{1/h_n}=1$, and this tells that this ratio is bounded when $n$ is large. Do you agree?

I'm unable to show this limit though.

Answer 1

$h_{n}\rightarrow 0$ so $1/h_{n}\rightarrow\infty$, so large $n$, $1/h_{n}>1$. Anyway, we have $\lceil u\rceil\leq 1+u$, so $\lceil 1/h_{n}\rceil\leq 1+1/h_{n}<1/h_{n}+1/h_{n}=2/h_{n}$ for large $n$.