Prove existance of positive step size for descent direction

Question

How can we prove that there exists positive step size t that quarantiees: f(x) > f(x + td), if d is descent direction of some cost function f : R n → R , such that ∇ f ( x ) d< 0 > ?

Answer 1

First, show that $$\nabla f \cdot \mathbf d = \lim_{t\to 0}\frac{f(\mathbf x+t\mathbf d) - f(\mathbf x)}t$$

The limit can only be negative if $f(\mathbf x+t\mathbf d) - f(\mathbf x) < 0$ for small $t$.