statement, assume the existence of $c$ as in the statement of the theorem. Let $v_i=i(c(1/i)-p)$ for $i=1,2,\dots$. Since $c(1/i)$ and $p$ are both contained in $C$, $v_i$ is in the tangent cone to $C$ at $p$. Moreover, $$v=\lim_{t\to0+}\frac{c(t)-p}{t}=\lim_{i\to+\infty}\frac{c(1/i)-p}{1/i}=\lim_{i\to+\infty}v_i.$$ So $v$ is in the sequential closure of the tangent cone to $C$ at $p$.

We observe the following reformulation of Theorem \[thm:characterization\] that is appropriate for normed spaces.

\[cor:normed\] Assume that $V$ is a normed space, that $p$ is an element of a closed, convex subset $C$ of $V$, and let $v\in V$. Then there is a continuous curve $c\colon[0,+\infty)\to C$ such that $c(0)=p$ and $dc(t)/dt|_{t=0+}=v$ if, and only if, $\theta(v){\geqslant}0$ for all $\theta\in V^*$ for which $\theta(p)=\inf_{q\in C}\theta(q)$.

This follows easily from Theorem \[thm:characterization\] since in this context the topological closure coincides with the sequential closure of any subset of $V$.

Here is the reformulation of the result that we will actually use in the following section.

\[cor:useful\] Let $V$ be a locally-convex topological vector space, $p$ an element of a convex subset $C$ of $V$, and $v$ be an element of $V$. Assume that $v$ and $C$ are contained in a subset $X$ of $V$ such that there is 