$\displaystyle X\subset\mathbb{R}^n,x\in\overline{X},v\in\mathbb{ R}^n$ so that $\displaystyle \lim_{s\to0+}\frac{1}{s}\mathrm{d}(x+sv,X)=0.$

I have to show that $\displaystyle \forall s_k\to0+,\exists x_k\to x$ so that $\displaystyle x_k\in X,\forall k$ and $\displaystyle \frac{x_k-x}{s_k}\to v.$

The first line means that $\displaystyle v$ belongs to the "adjent cone" to $\displaystyle X$ at $\displaystyle x$ and there are several equivalent definitions. This implication is the only one I cannot prove.

Thanks, AMI