Let $f$ be Gâteaux differentiable and a function of a real variable, show it is Fréchet differentiable.

Question

Let $\mathbb{Y}$ be a normed space and $x \in \mathbb{R}$. A function $f: \mathbb{R} \to \mathbb{Y}$ is Gâteaux differentiable. Show it is also Fréchet differentiable at $x$.

We have that Fréchet implies Gâteaux, and I know that the converse isn't necessarily true; however I believe it is true in the case of a function of a real variable but cannot show it rigorously.