Uniqueness of multivariable derivative

Question

I've found an answer of the question in Proof that the derivative is unique?, but I don't quite understand the answer, how the proof uses triangle inequality? And how it use $\alpha y \rightarrow 0$ and linearity of $\sigma_{1}'-\sigma_{2}'$ to get the equation below? Thanks very much!

Answer 1

$$\|\frac{(\sigma_2'-\sigma_1')(y)}{y}\| \leq \frac{\|(\sigma_2'-\sigma_1')(y)\|}{\|y\|} \leq \frac{\|\sigma(x+y)-\sigma(y)-\sigma_1'(y)\|}{\|y \|}+ \frac{\|\sigma(x+y)-\sigma(y)-\sigma_2'(y) \|}{\| y\|}$$

Since norms are non-negative, the limit when $\|y\|$ goes to $0$ is equal to $0$. As for the second question, linearity means we can pull out $\alpha$ from both numerator and denominator, so they cancel.