# How to prove every Cauchy Sequence in $\mathbb{R}^n$ converges

This is an analysis exercise that I have been struggling with for some time now. I am not familiar with metric spaces.

In $$\mathbb{R}$$, the book that I am using proves this fact by showing that every Cauchy sequence in $$\mathbb{R}$$ is bounded. Next, they use Bolzano-Weierstrass to choose a convergent subsequence of that Cauchy sequence.

However, the book does not specify an analog to boundedness in $$\mathbb{R}^{n}$$. Also, the book proved Bolzano-Weierstrass for $$\mathbb{R}$$, not $$\mathbb{R}^{n}$$. I was originally planning to outline the $$\mathbb{R}$$ approach by proving boundedness and choosing a convergent subsequence, but this is not currently possible because of what I said.

I was wondering if there is a good way to do this problem.

Thanks

• Apply the result in $\Bbb R$ to prove the result for $\Bbb R^n$. – Lord Shark the Unknown Feb 6 at 4:46
• To supplement the comment above, show that you can apply the result in $\Bbb R$ to each component of your sequence to show that each component of the sequence converges. Use that fact (and the fact that you're in a finite-dimensional space) to show that the sequence itself converges. – Robert Shore Feb 6 at 5:11

So, first, we need a distance on $$\mathbb{R}^k$$. We have a number of choices; the usual Euclidean distance $$d_2(x,y)=\sqrt{\sum_j (x_j-y_j)^2}$$, the "taxicab" distance $$d_1(x,y)=\sum_j |x_j-y_j|$$, the sup norm $$d_{\infty}(x,y)=\max_j |x_j-y_j|$$, and others. (In these definitions, $$x_j$$ and $$y_j$$ are the components of $$x$$ and $$y$$ respectively.)

Once we have a distance, we can define bounded sets, Cauchy sequences, and convergence in exactly the same way we did for $$\mathbb{R}$$:

• A set $$S$$ is bounded if there is some $$R\in \mathbb{R}$$ and $$y\in \mathbb{R}^k$$ such that $$d(y,x) < R$$ for all $$x\in S$$.
• A sequence $$x(n)$$ is Cauchy if for all $$\epsilon>0$$, there is an $$N$$ such that $$d(x(n),x(m)) < \epsilon$$ whenever $$m,n > N$$.
• A sequence $$x(n)$$ converges to $$x$$ if for all $$\epsilon>0$$, there is an $$N$$ such that $$d(x_n,x) < \epsilon$$ whenever $$n>N$$.

This is all general stuff for the topic of metric spaces - standard definitions.

So, then, what about $$\mathbb{R}^k$$? As it turns out, our choice of distance doesn't really matter; within a pretty broad class that includes all the example I gave, they're "equivalent". The exact distances between a pair of points may vary, but we can bracket a distance function of a pair of points between two constant multiples of another distance function. For example, $$d_{\infty}(x,y) \le d_1(x,y)\le k\cdot d_{\infty}(x,y)$$ for any $$x,y$$.
Because of this property, any of these distance functions induce the same topology - the same bounded sets, Cauchy sequences, convergent sequences, open sets, and closed sets.

And with that, a theorem: Under any of these "nice" distances, a set $$S$$ is bounded if and only if each of the component sets $$S_i=\{x_i: x\in S\}$$ is bounded. A sequence $$x(n)$$ converges to $$x$$ if and only if each of the component sequences $$x_i(n)$$ converges to $$x_i$$. A sequence $$x(n)$$ is Cauchy if and only if each of the component sequences $$x_i(n)$$ is Cauchy.

With that, we can prove Bolzano-Weierstrass in $$\mathbb{R}^k$$ by applying the version from $$\mathbb{R}$$. Start with a bounded sequence in $$\mathbb{R}^k$$. The first components are bounded, so we extract a subsequence with convergent first components from that. Then the second components are bounded, so we extract a further subsequence with convergent second components - and its first components still converge. Repeat this process $$k$$ times to reach a subsequence with each of its components convergent, and we have it.

I am not familiar with metric spaces.

You'll need to be. Simply defining things like bounded sequences and Cauchy sequences requires that distance function. Convergence gives you a choice - either metric spaces or more general point-set topology. The metric spaces are usually treated earlier, because they're more familiar.

Fortunately, these topics will come up in your course, or possibly a later course in the sequence.