Am I right in saying that the unit ball is convex if for any 2 points in the unit ball, the straight line segment between them is also in the unit ball?

The book uses Cauchy-Schwarz to prove convexity of the ball. Then they do the following:

$\displaystyle s^{2}|x|^2+ 2st|x||y| + t^{2}|y|^2 \leq (s+t)^2 = 1 $ hence $\displaystyle |z | \leq 1 $ which proves convexity of the ball.

How did they deduce the above inequality?

$\displaystyle z = sx+ty $ where $\displaystyle 0 \leq s,t \leq 1 $.