Theorem 3.2 (d) of Introduction to Analytic Number Theory by Apostol states
$$ \sum_{n \le x} n^a = \frac{x^{\alpha + 1}}{\alpha + 1} + O(x^\alpha) \text{ if } \alpha \ge 0.$$
Then later it uses this theorem in Theorem 3.3 it uses Theorem 3.2 (d) in this way in a specific step:
Now we use part (d) of Theroem 3.2 with $ \alpha = 0 $ to obtain $$ \sum_{q \le x/d} 1 = \frac{x}{d} + O(1).$$
The entire Theorem 3.3 or what it is, is not important for this question. This question is about only the step quoted above.
Isn't invoking Theorem 3.2 (d) too heavy to prove just $ \sum_{q \le x/d} 1 = \frac{x}{d} + O(1)? $ I think I can prove it with elementary arithmetic like this:
$$ \sum_{q \le x/d} 1 = \left\lfloor \frac{x}{d} \right\rfloor = \frac{x}{d} - \left(\frac{x}{d} - \left\lfloor \frac{x}{d} \right\rfloor\right) = \frac{x}{d} + O(1). $$
Why invoke a heavy theorem like Theorem 3.2 (d) when it can be proven with much less? Or am I missing something?