So imagine I had a new number type. The idea is simple: It is an array (possibly with infinitely many elements) that you can do addition on it. For example, {1,2}+3 and {1,2}+{3,4} would be a valid thing. Note, however, that order is important, and that {1,2}+{3,4} is not equal to {3,4}+{1,2}. I call these numbers "ordinals".

I then have a function. It takes an ordinal and gives an output through some algorithm. I will be using this function:

1) f(0,n) = n
2) f(ß+1,n) = f(ß,n+1)
3) f(ß,n) = f(ß[n],n)

Let $(V,\mathbb K,\langle{.,.}\rangle)$ be an inner product space with basis $\gamma=\langle c_1,\dots,c_n\rangle$. Consider the nested subspaces $U_1\subset\dots\subset U_n$ induced by $\gamma$, where $U_k=\operatorname{span}\{c_1,\dots,c_k\}$. Then there exists an orthonormal basis $\beta$ that induces the same nested subspaces.

The proof starts as follows:

Define: $U_0=\{0\}$. Voor each $k\in\{1,\dots,n\}$, choose $b_k\in U_k$, such that $b_k\perp U_{k-1}$ and $\Vert b_k\Vert=1$.

This not equivalent to saying: chose $b_k\in U_k$, such $b_k=\lambda c_k$ (for some $\lambda\in\mathbb K$)…