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I understand this is not a research level question but I really want to know, would anyone please help. This question is related to the signatures that arises in rough path theory. https://en....

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...