A reference for some fact in analysis

I am looking for a reference for the following fact. Any hints would be appreciated.

Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ we have

$\sum_{k=1}^\infty a_k f(x_k) = \sum_{k=1}^\infty a_k f(y_k)$.

Then

$x_k = y_k$ for $k\in \mathbb{N}$.

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It's not true. For example, if some $a_j = a_k$, you could take $(y_n)$ to be $(x_n)$ with $x_j$ and $x_k$ interchanged.