I am reading the proof of the following lemma:

Let $\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}$ be differentiable at $\displaystyle \bar{\mathbf{x}}}$. If $\displaystyle \nabla f(\bar{\mathbf{x}})^{\mathrm{T}} \mathbf{d} < 0$, then $\displaystyle \mathbf{d}$ is a decent direction for $\displaystyle f$ at $\displaystyle \bar{\mathbf{x}}}$.

I don't understand the first line of the proof, which states: because $\displaystyle f$ is differentiable at $\displaystyle \bar{\mathbf{x}}}$ then

$\displaystyle f(\bar{\mathbf{x}} + \lambda \mathbf{d}) = f(\bar{\mathbf{x}}) + \lambda\nabla f(\bar{\mathbf{x}})^{\mathrm{T}} \mathbf{d} + \lambda ||\mathbf{d}||\alpha (\lambda \mathbf{d}) $

where

$\displaystyle \lim_{\lambda \rightarrow 0}\alpha (\lambda \mathbf{d}) = 0$

They define a decent direction as:

$\displaystyle \exists \delta > 0: f(\bar{\mathbf{x}} + \lambda \mathbf{d}) < f(\bar{\mathbf{x}}) \; \forall \lambda \in (0, \delta)$

I don't understand what $\displaystyle \alpha$ is exactly. Is it just the Fréchet derivative?