Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

Question

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy?

I've proven that the triangle inequality holds for the euclidean norm of vectors.

Answer 1

Like 900 sit-ups a day pointed out a sequence $(x_k)_{k\in\mathbb{N}}$ in $\mathbb{R}^n$ converges if and only if the sequences $(\pi_i(x_k))_{k\in\mathbb{N}}$ converge for all $i$ with $1\le i\le n$ where $\pi_i:\mathbb{R}^n\to\mathbb{R}$ is the projection to the $i$th coordinate. So you have to show that the sequences $(\pi_i(x_k))_{k\in\mathbb{N}}$ are Cauchy and then use the result for $\mathbb{R}$ that a sequence converges if and only if it is Cauchy.