I am trying to prove or disprove the following statement:
For a general inner product space $(V, \langle\cdot,\cdot\rangle)$, if a sequence $(\vec{w}_n)_n$ in $V$ is such that $\lim_{n\to\infty} (\vec{w}_{n+1}-\vec{w}_n) = \vec{w}_1$, then the orthonormal set $({\vec{v}_n})_n$ obtained from $(\vec{w}_n)_n$ through the Gram-Schmidt process also satisfy the property that $\lim_{n\to\infty} (\vec{v}_{n+1}-\vec{v}_n) = \vec{v}_1$.
What I have attempted so far:
\begin{align*} \lim_{n\to\infty} (\vec{v}_{n+1}-\vec{v}_n) &= \lim_{n\to\infty}(\vec{w}_{n+1}-\sum_{j=1}^n\langle\vec{v}_j, \vec{w}_{n+1}\rangle\vec{v}_j - \vec{w}_n + \sum_{j=1}^{n-1}\langle\vec{v}_j, \vec{w}_{n}\rangle\vec{v}_j)\\ &= \lim_{n\to\infty}(\vec{w}_{n+1}-\vec{w}_n) - \lim_{n\to\infty}(\sum_{j=1}^n\langle\vec{v}_j, \vec{w}_{n+1}\rangle\vec{v}_j - \sum_{j=1}^{n-1}\langle\vec{v}_j, \vec{w}_{n}\rangle\vec{v}_j)\\ &= \vec{w}_1 - \lim_{n\to\infty}(\sum_{j=1}^n \langle\vec{v}_j, \vec{w}_{n+1}-\vec{w}_n\rangle\vec{v}_j + \langle\vec{v}_n,\vec{w}_n\rangle\vec{v}_n) \end{align*} I'm not sure where to go from here...