Gradient descent is based on the observation that if the multi-variable function $$F(x)$$ is defined and differentiable in a neighborhood of a point $$a$$ , then $$F(x)$$ decreases fastest if one goes from $$a$$ in the direction of the negative gradient of $$F$$ at $$a$$, $$-\nabla F(a )$$. It follows that, if
$$a_{n+1}=a_n-\gamma \nabla F(a_n)$$
for positive $$\gamma$$ that is small enough, then $$F(a_n) \ge F(a_{n+1})$$.