I'm currently taking my algorithms class, and I learnt that Big theta is defined as follows:
$f(n) = \Theta(g(n))$ if there exist constants $c_1, c_2, n_0 > 0$ such that $0 ≤ c_1g(n) ≤ f(n) ≤ c_2g(n)$ for all $n ≥ n_0$.
So if our $f(n)$ and $g(n)$ are of the same order, say $n$ and $2n$, we can easily find a constant $c$ and an $n_0$, such that $n$ lies between $c_1(2n)$ and $c_2(2n)$ for all $n ≥ n_0$.
Suppose in contrast that our $f(n)$ and $g(n)$ are of different order, say $f(n)$ is $O(g(n))$ where $f(n) = n$ and $g(n) = n^2$. So technically, there will exist constants $c_1$ and $c_2$ such that $c_1(n^2)$ is less than $n$, and $c_2(n^2)$ is more than $n$. However, because $n^2$ is of a higher order than $n$, as $n$ increases, $n^2$ will surpass n eventually regardless of its constant, and thus the definition of $\Theta(n)$ where for all $n \ge n_0$ will never hold if our $g(n)$ and $f(n)$ eventually intersect each other (which will happen if they have different orders?).
Is this always the case? Or am I being too narrow-minded in thinking like this.