big theta question,not sure is book mistake or my misunderstand.... I have $\Theta$-notation question from my new book example solution... the example question is:

Find $\Theta$ bound for $$f(n)={n^2\over 2} - {n\over 2}$$

and the book solution is :

$\displaystyle {n^2\over5} \le {n^2\over 2} - {n\over 2} \le n^2$, for all $n\ge 1$
so $\displaystyle{n^2\over2} - {n\over2} = \theta(n^2)$ with $c_1=\frac15$, $c_2=1$ and $n_0 = 1$.

what I don't understand is why lower bound is $\frac15$ and $n_0$ is $1$, if I put $1$ to formula it will be $\frac15 \leq 0 \leq 1$, it's not correct...

I thought the answer would be

$\displaystyle{n^2\over 4} \leq {n^2\over 2} - {n\over2} \leq n^2$, for all $n\geq2$
so $c_1=\frac14$, $c_2=1$, $n_0=2$.
could somebody explain why?
It is a typo: please put n0=2.

It is valid with n0=2
It is a typo: please put n0=2.

It is valid with n0=2