Useful Tools
Triangle Inequality
For any norm $\|\cdot\|$ , the following inequation holds:
$$\|\boldsymbol{a} + \boldsymbol{b}\| \le \|\boldsymbol{a}\| + \|\boldsymbol{b}\|$$
Cauchy-Schwartz Inequality
$$\left(\boldsymbol{x}^\mathsf{T}\boldsymbol{y}\right)^2 \le \boldsymbol{x}^\mathsf{T}\boldsymbol{x}\; \cdot \;\boldsymbol{y}^\mathsf{T}\boldsymbol{y}$$
$$|\boldsymbol{x}^\mathsf{T}\boldsymbol{y}| \le \|\boldsymbol{x}\|_2 \cdot \|\boldsymbol{y}\|_2.$$