I've been trying to answer the same question answered here: Second derivative "formula derivation"

And I'm stuck in a step that is not addressed both in the answer and in the comments of the question over there. In the original question he uses the fact that

$$f''(x) = \lim_{h\to0} \frac{f'(x+h) - f'(x)}{h}$$ $$f''(x) = \lim_{h\to0} \frac{ \frac{ f(x+2h) - f(x+h)}{h} - \frac{ f(x+h) - f(x)}{h} }{h}$$

Which I basically see as taking the derivatives with the same limit 3 times. Shouldn't it be as follow?

$$f''(x) = \lim_{h\to0} \frac{ \lim_{h_1\to0}\frac{ f(x+h+h_1) - f(x+h)}{h_1} - \lim_{h_2\to0}\frac{ f(x+h_2) - f(x)}{h_2} }{h}$$

How do you justify moving to the equation given in the original answer?