1. The problem is to verify whether a given function is a metric on $\mathbb{R}^n$. 2. A function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is a metric if it satisfies the following properties for all $x,y,z \in \mathbb{R}^n$: - (Non-negativity) $d(x,y) \geq 0$ - (Identity of indiscernibles) $d(x,y) = 0$ if and only if $x = y$ - (Symmetry) $d(x,y) = d(y,x)$ - (Triangle inequality) $d(x,z) \leq d(x,y) + d(y,z)$ 3. To verify a specific function is a metric, check each property step-by-step. 4. Since the user did not specify the function, let's consider the standard Euclidean distance on $\mathbb{R}^n$: $$d(x,y) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}$$ 5. Check non-negativity: Since squares are non-negative and square root is non-negative, $d(x,y) \geq 0$. 6. Check identity of indiscernibles: $d(x,y) = 0$ means $\sum (x_i - y_i)^2 = 0$, which implies $x_i = y_i$ for all $i$, so $x = y$. 7. Check symmetry: $d(x,y) = \sqrt{\sum (x_i - y_i)^2} = \sqrt{\sum (y_i - x_i)^2} = d(y,x)$. 8. Check triangle inequality: By Minkowski inequality, $d(x,z) \leq d(x,y) + d(y,z)$. 9. Therefore, the Euclidean distance is a metric on $\mathbb{R}^n$. If you provide a specific function, I can verify it accordingly.