Cauchy sequences through examples in Quantum Mechanics (at the level of the rigor of physicists) I have just read the definition of a Cauchy sequence:

A sequence ($\psi_n$) is a Cauchy sequence in a vector space $V$ when $||\psi_n-\psi_m||\to 0$ when $n,m\to\infty$. The limit of every Cauchy sequence $(\psi_n)$ converges to a definite element $\psi\in V$ i.e. $$\lim\limits_{n\to\infty}\psi_n=\psi.$$

But I cannot feel it completely unless I see an example of such a sequence. What is an example of a Cauchy sequence of vectors $(\psi_1,\psi_2,...)$ that we encounter in quantum mechanics?
 A: Here's a concrete example. For a particle in an infinite potential well of width $a$, the normalized energy eigenvectors are of the form
$$\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$$
Most wavefunctions - such as the $\Psi(x) = \frac{1}{\sqrt{a}}$, corresponding to a uniform spatial probability density throughout the well - cannot be written as a finite linear combination of energy eigenvectors.  It can, however, be expressed as the sum of the convergent series
$$ \sum_{n=1}^\infty\frac{2\sqrt{2}}{(2n-1)\pi} \psi_{2n-1} \rightarrow \Psi(x)$$
as illustrated with the following plot of the first $n$ partial sums:

The sequence of partial sums $\Psi^{(n)}:= \sum_{k=1}^n\frac{2\sqrt{2}}{(2n-1)\pi}\psi_{2n-1}$ is indeed Cauchy (which can be verified as a nice exercise), as requested.


I have just read the definition of a Cauchy sequence [...]

Note that the proper definition of a Cauchy sequence is that, for any $\epsilon>0$, there exists some $N\in \mathbb N$ such that for all $n,m>N$, $\Vert \psi_n-\psi_m\Vert <\epsilon$. In words, given any arbitrarily small tolerance $\epsilon$, if we go far enough along in the sequence we can find a point $N$ past which all of the terms from the $N^{th}$ onward are within $\epsilon$ of one another.
The definition you provide is problematic, in my opinion. In what way do $n,m\rightarrow \infty$?  Do you mean that we hold $n$ fixed, send $m\rightarrow \infty$, and then send $n\rightarrow \infty$ afterward? If that's the case, then we need the sequence to have a well-defined limit which is a priori not necessary for a generic Cauchy sequence.  Or we send them both to infinity at the same time? If that's the case, it matters how exactly we do this.
A: The most common way Cauchy sequence appear is as series. For example, if $|0\rangle, |1\rangle, \ldots ,|n\rangle , \ldots$ is an orthonormal sequence and $\sum |a_n|^2 = 1$, we expect $|\psi \rangle = \sum a_n |n\rangle$ to be a well defined state, ie we would like the limit :
$$\lim_{N\to +\infty} \sum_{n=0}^N a_n|n\rangle$$
to exist.
The sequence $(\sum_{n=0}^N a_n|n\rangle)_{N\in\mathbb N}$ is Cauchy, since :
$$\left\|\sum_{n=0}^{N+p}a_n|n\rangle - \sum_{n=0}^{N}a_n|n\rangle \right\|^2 = \left\| \sum_{n=N+1}^{N+p}a_n|n\rangle\right\|^2 = \sum_{n=N+1}^{N+p}|a_n|^2\overset{N\to +\infty}{\longrightarrow}0$$
Therefore, because we work in a (complete) Hilbert space, the limit we wanted does indeed exist.
