1. Let's state the problem: You have a 2D vector \(\mathbf{v}_n = (x_n, y_n)\) that converges to \((0,0)\). 2. The question is whether \(x_n\) and \(y_n\) converge together or separately. 3. By definition, a vector \(\mathbf{v}_n = (x_n, y_n)\) converges to \((0,0)\) if and only if both components converge to their respective limits: $$\lim_{n \to \infty} x_n = 0 \quad \text{and} \quad \lim_{n \to \infty} y_n = 0$$ 4. This means the convergence of the vector is equivalent to the convergence of each component separately. 5. Therefore, \(x_n\) and \(y_n\) converge separately to 0, and together they form the vector convergence to \((0,0)\). 6. In plain language: The vector converges to \((0,0)\) only if both the \(x\)-coordinate and the \(y\)-coordinate individually get closer and closer to 0 as \(n\) increases. Final answer: \(x_n\) and \(y_n\) converge separately to 0, and together this means the vector converges to \((0,0)\).