Orders of convergence

Hi Guys,

I have a small question on orders of convergence.
I'm happy with the idea that for $N\in\mathbb{N}$ if,
$\text{error} \le C_1N^{-\alpha}+C_2N^{-\beta}$ , where $\alpha<\beta$ , $\alpha, \beta \in\mathbb{R}$ , $\alpha,\beta>0$ , $C_1, C_2 \in\mathbb{R}$ are constants
$\Rightarrow \text{error} \le (C_1+C_2)N^{-\alpha}$

or

$\text{error} \le C_1N^{\alpha}+C_2N^{\beta}$ , where $\alpha<\beta$ , $\alpha, \beta \in\mathbb{R}$ , $\alpha,\beta>0$ , $C_1, C_2 \in\mathbb{R}$ are constants
$\Rightarrow \text{error} \le (C_1+C_2)N^{\beta}$

but for
$\text{error} \le C_1N^{-\alpha}+C_2N^{\beta}$ , where $\alpha, \beta \in\mathbb{R}$ , $\alpha,\beta>0$ , $C_1, C_2 \in\mathbb{R}$ are constants.
is this the right outcome
$\text{error} \le (C_1+C_2)N^{\beta}$ ?
Other than when 0 < N < 1 is there a time when $N^{-\alpha}$ would win?
Thanks :-)