Proof of Convergence of the Sum of Components in a Hilbert Space I recently started studying the fascinating mathematical structures of Hilbert spaces. As a physics guy, I worked with Hilbert spaces in quantum mechanics without knowing the rigorous definition of what they are.
I was studying the completeness of the Hilbert space and a basis of such space.
I am supposed to prove the following:
$$
\text{If } \{ |e_i\rangle \} \text{ is an infinite sequence of orthonormal kets in } \mathscr{H} \text{, then for any ket } |f\rangle \in \mathscr{H},\\
\text{the series }\sum_{i=1}^\infty {|e_i\rangle \langle e_i|f \rangle} \text{ converges in } \mathscr{H}.
$$
My attempt was to define a sequence of kets $| a_n \rangle := \sum_{i=1}^{n}{|e_i\rangle \langle e_i|f \rangle}$ and to prove that it is a Cauchy sequence, i.e., the norm $\lVert |a_i\rangle - |a_j\rangle \rVert$ gets arbitrarily close to zero for sufficiently large $i, j$. However, I can't seem to grasp how this should be done, especially since the completeness of the chosen basis is not assumed. Any hint would be appreciated. Thanks.

PS: In case there are many conventions of the definition of Hilbert spaces, I state the version I am working with.

*

*The Hilbert space $\mathscr{H}$ vector space over the field of complex numbers $\mathbb{C}$.

*It has an inner product $\left< \cdot, \cdot \right>$ which defines the norm $\lVert \cdot \rVert$ on the space.

*It is complete, i.e., every Cauchy sequence in $\mathscr{H}$ converges to a ket in $\mathscr{H}$.

 A: [Note that I drop the bra-ket notation in my answer, as I'm not familiar with it]
You have the right idea : you need to prove that $(a_n)_{n\ge1}$ is a Cauchy sequence. The proof of that fact relies on the orthonormality of the family $(e_k)_{k\ge 1}$ : w.l.o.g. assume $1\le i<j$, then we have
$$\begin{align*}\|a_i - a_j\|^2 &= \left\|\sum_{k=1}^i\langle e_k,f\rangle e_k - \sum_{k=1}^j\langle e_k,f\rangle e_k\right\|^2\\
&=\left\|\sum_{k=i+1}^j\langle e_k,f\rangle e_k\right\|^2\\
&=\sum_{k=i+1}^j|\langle e_k,f\rangle|^2\end{align*} $$
Where the last equality follows from orthonormality.
All we need now is to prove that $S_n := \sum_{k=1}^n|\langle e_k,f\rangle|^2$ converges to a real number as $n\to\infty$ to conclude that $(a_n)_{n\ge 1}$ is Cauchy. The convergence of this sum follows from Bessel's inequality, which is proven as follows :
$$\begin{align*}0\le \left\|f - \sum_{k=1}^n\langle e_k,f\rangle e_k \right\|^2 &= \|f\|^2 - 2\sum_{k=1}^n\langle e_k,f\rangle\langle e_k,f\rangle + S_n\\
&= \|f\|^2 - S_n  \end{align*} $$
Hence we have that for all $n$, $S_n\le \|f\|^2$ which implies that $\lim_\limits{n\to\infty}S_n <\infty $ and $(a_n)_{n\ge 1}$ is Cauchy, as desired.
