Norm squared inequality

At first I thought it was just the triangle inequality and adding/subtracting $c$ inside the the first norm.

That, and then from the obtained

$$\lVert a-b\rVert^2 \leqslant \lVert a-c\rVert^2 + 2\lVert a-c\rVert\cdot\lVert c-b\rVert + \lVert c-b\rVert^2$$

we conclude using the inequality

$$2uv \leqslant u^2 + v^2$$

for $u, v \in \mathbb{R}$, following from $(u-v)^2 \geqslant 0$.