Let $f_k$ be a real numbers such that $\sum_{k=1}^\infty f_k < \infty$. For each $R > 0$, define the convergent sum $$v(R) = \sum_{k=1}^\infty f_k(b_k(R)e^{-ky} - c_k(R)e^{ky})$$ where $0 \leq y \leq R$ and where $b_k$ and $c_k$ are functions of $k$ and $R$ and satisfy $$b_k(R) = 1+ \frac{1}{e^{2kR}-1}\quad \text{and}\quad c_k(R) = b_k(R)-1 = \frac{1}{e^{2kR}-1}$$
Apparently this sum uniformly converges (see this) in $n$ as function of $R$. How do I show this? I.e. if $v_n(R) = \sum_{k=1}^n f_k(b_k(R)e^{-ky} - c_k(R)e^{ky})$ how do I show that $v_n \to v$ uniformly as function of $R$?
I didn't understand how the explanation is given in that thread and I think my question is too basic to be asked there. Both the coefficients have look "like" $e^{-2kR}$ so it seems like it should be uniform but I don't know the argument.