We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Let $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$