= Solution
For $g=\nabla f(x)\ne0$, the unit <steepest descent direction> is $-g/\|g\|_2$ by the <Cauchy-Schwarz inequality>. The usual unnormalized direction is $\boxed{p=-g}$; choosing its scale separately as a line-search step gives the same ray. It satisfies $\boxed{g^Tp=-\|g\|_2^2<0}$, so it is a <descent direction>. The first-order <Taylor expansion> then guarantees decrease for sufficiently small positive steps. The subsequent bounds use this unnormalized negative-gradient convention.
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