In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set
$$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
$\mathbf{6}$ ▷ Prove the inequalities
$$\forall k \in \mathbf{N}^* \quad \left\| X_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right) \text{ and } \left\| Y_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right).$$