Determine if the sequence $x_k \in \mathbb{R}^3$ is convergent when $x_k=(2, -k^{-1}, k^{-3})$

Question

Determine if the sequence $x_k \in \mathbb{R}^3$ is convergent when $$x_k=(2, -k^{-1}, k^{-3})$$

I remember seeing a theorem that stated that the convergence of the coordinates would satsify the convergence of the sequence (please correct me if I'm wrong.)?

If this is the case it's quite clear that $k^{-1}, k^{-3} \to 0$ as $k \to \infty.$

However I'm not sure what would be my choice for $a$ when looking at $$||x_k-a|| = ||(2,-k^{-1}, k^{-3})-a||$$

I know that $a$ should be the limiting value if the sequence converges, but here it's not stated that what would it converge to if it converges.

Answer 1

As you say, $k^{-1}$ and $k^{-3}$ both tend to zero, and to take the limit of a vector, you can just take the limits of the components. So your $a$ should be $(2,0,0)$.

What you do next depends on your class. You could simply apply the theorem to conclude immediately that the sequence $x_k$ converges to $(2,0,0)$, or you could prove it formally, for example by using an $\epsilon$-type argument.