Let $S$ be a subset of $X$ and $X$ is a normed space. The tangent cone to $S$ at a point $x \in S$, denoted $T_S(x)$, defined by $$ T_S(x)=\{v:\exists \ t_k \downarrow 0, \exists\ v_k \to v \text{ such that } \ x+t_k v_{k} \in S\} $$ and normal cone $N_{S}(x)$ is defined by $$ N_{S}(x)=\{\xi \in X^{*}:\left<\xi,v\right>\leq 0 \ \forall v \in T_{S}(x)\}. $$ Then can deduce that $$ x\in \text{int} \ S \ \Longrightarrow T_{S}(x)=X, \ N_{S}(x)=\{0\}, $$ where $\text{int} \ S$ denotes the the set of interior point of $S$.
My question is how to prove it.