In this proof, I'm not clear with two things:
1) how can we break $$\lim_{\mathbf x \to \mathbf a}\|{\mathbf f(\mathbf x)-\mathbf f(\mathbf a)-L(\mathbf x-\mathbf a)}\| +\|L(\mathbf x-\mathbf a)\|$$
into
$$\lim_{\mathbf x \to \mathbf a}\|{\mathbf f(\mathbf x)-\mathbf f(\mathbf a)-L(\mathbf x-\mathbf a)}\| + \lim_{\mathbf x \to \mathbf a}\|L(\mathbf x-\mathbf a)\|$$ without knowing beforehand that $$\lim_{\mathbf x \to \mathbf a}\|L(\mathbf x-\mathbf a)\|$$ exists?
2) If that is the case, how do we know that $$\lim_{\mathbf x \to \mathbf a}\|L(\mathbf x-\mathbf a)\| =0 ?$$
Any help appreciated.
Note that since $L$ is linear then $$\lim_{x\to a}\|L(x-a)\|=\lim_{x\to a}\|L(x)-L(a)\|=\|L(a)-L(a)\|.$$