Prove that if $A_n \rightarrow \infty$ and $B_n \rightarrow \infty$ then (a) $A_n+B_n \rightarrow \infty$

Question

Let $A > 0$ be given. There exists a natural number $N_1$ Such that $a_n > A$ for all $n > N_1$.

Let $A > 0$ be given. There exists a natural number $N_2$ Such that $b_n > A $for all $n >N_2$.

Now let $N = \max\{N_1,N_2\}$. For all $n > N$, $a_n$ + $b_n > 2A$ by previous steps $> A$ as required.

My question is would it have been better to let $A = \frac{A}{2}$ as I would deduce that the sequence $> A$ straight away. Basically, am I wrong for doing what I did ?