I was learning ϴ(n) notation in my course "Asymptotic Analysis for Algorithms" when I encountered the following example:
For any non-negative constants $c_1\geq 0,c_2\geq 0,n\geq n_0$ we have the following inequality:
$$c_1\leq\frac{1}{2}-\frac{3}{n}\leq c_2$$
For these constants to satisfy this inequality, will be $c_2\geq\frac{1}{2}$ when $n\geq 1$. This is the part I'm having issues with.
I tried to derive it assuming $n \geq 1$ and I found:
\begin{align*} c_2 &\geq \frac{1}{2} - \frac{3}{n} \\ \implies c_2 &\geq \frac{1}{2} - \frac{3}{1} \\ \implies c_2 &\geq -\frac{5}{2}` \end{align*}
I managed to show the left inequality holds when $n\geq 7$ and $c_1\leq \frac{1}{14}$.
The point is $-3/n$ is increasing to zero, so for large $n$ $1/2-3/n$ is approximately (but less than) $1/2$
In other words $1/2-3/n \ge 1/2-3/1$ is wrong, so you cannot derive the second inequality from the first one.