Steps of Verify Completeness of a Normed Vector Space

Question

The textbook by Kreyszig on functional analysis identifies three steps to verify the completeness of a space:

1. Construct an element to use as a limit: $x$
2. Prove that $x$ is in the space considered
3. Prove that $x_n \rightarrow x$ (in the sense of metric)

I'm confused by these steps. In particular, if we stop at step 2, aren't we done? If $x$ is a limit point in a metric space (in fact a normed space here), aren't we gauranteed that a sequence hits it? (Take balls of radius $1/n$)

Answer 1

Completeness means every Cauchy sequence converges. More precisely, every sequence $(x_n)$ satisfying $\|x_n-x_m\| \rightarrow 0$ as $n,m\rightarrow \infty$, converges$^3$, to some point$^1$, in the space$^2$. You need point 3 because of the definition of convergence in metric spaces.

What if you took the sequence $1/n$ in $[0,1]$? I could construct the candidate limit "1". I could show that 1 belongs to the space. But I couldn't show convergence of the sequence to 1.

Answer 2

Here's the full context (excerpted from the text):

In various applications a set $X$ is given (for instance, a set of sequences or a set of functions), and $X$ is made into a metric space. This we do by choosing a metric $d$ on $X$. The remaining task is then to find out whether $(X, d)$ has the desirable property of being complete. To prove completeness, we take an arbitrary Cauchy sequence $(x_n)$ in $X$ and show that it converges in $X$. For different spaces, such proofs may vary in complexity, but they have approximately the same general pattern:

(i) Construct an element $x$ (to be used as a limit).

(ii) Prove that $x$ is in the space considered.

(iii) Prove convergence $(x_n) \rightarrow x$ (in the sense of the metric).

Note:$\;$The point $x$ depends on the arbitrary Cauchy sequence $(x_n)$.

To prove that $X$ is complete, your goal is to show that an arbitrary Cauchy sequence $(x_n)$ in $X$ converges to some $x\in X$.

The author's suggested plan of attack:

• First you should try to figure out the right choice of $x$, based on some fixed, arbitrary Cauchy sequence $(x_n)$ in $X$.$\\[4pt]$
• Next verify that $x\in X$ (as opposed to $x$ in some metric space $Y$ containing $X$, but with $x\not\in X$).$\\[4pt]$
• Finally, assuming you have the right $x$, show that $(x_n)$ converges to $x$.

As regards your misconception . . .

You can't claim at the outset that $x$ is a limit point of the sequence $(x_n)$, since $x$ is not given. It's your job to find $x$ (for an arbitrary Cauchy sequence $(x_n)$), and to show that it works.