Question regarding Lipschitz condition $\| f_1 - f_2 \| \leq k \|x_1 - x_2\|$

Question

Consider global Lipschitz condition:

$\| f(x_1) - f(x_2) \| \leq k \|x_1 - x_2\|$

We can manipulate it to:

$\frac{\| f(x_1) - f(x_2) \|}{\|x_1 - x_2\|} \leq k $

But according to the definition of a norm, is it legal to then say the above is equivalent to: $\| \frac{f(x_1) - f(x_2) }{x_1 - x_2}\| \leq k $ ?

Which is equivalent to

$\| \frac{df}{dx}\| \leq k $ ?