1. ## Inequality proof

$\lVert \mathbf{u} + \mathbf{v} \rVert \geq \lVert \mathbf{u} \rVert - \lVert \mathbf{v} \rVert$

$\big(\lVert \mathbf{u} + \mathbf{v} \rVert \big)^2= <\mathbf{u} + \mathbf{v},\mathbf{u} + \mathbf{v}> = <\mathbf{v},\mathbf{v}> + 2<\mathbf{v},\mathbf{u}> + <\mathbf{u},\mathbf{u}>$

Cauchy-Schwarz: $<\mathbf{v},\mathbf{u}> \leq \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert$

$\lVert \mathbf{u} \rVert^2 + \lVert \mathbf{v} \rVert^2 + 2 \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert$

Not sure I am getting anywhere

2. Use the triangle inequality. $|y|=|y+x+(-x)| \le |y+x| + |x|$, which is what you want.