(related to Proposition: Calculations with Uniformly Convergent Functions)

By hypothesis, $\mathbb F$ is either the field of real numbers or the field of complex numbers, let $D\subset \mathbb F.$ Let $f_n,g_n,f,g:D\to\mathbb F$ be functions, $\alpha_n,\alpha\in\mathbb F,$ and let $f_n\to f,$ $g_n\to g$ are uniformly convergent, and $(\alpha_n)_{n\in\mathbb N}$ be a sequence with the limit $\alpha_n\to \alpha.$

Now, assume that $f_n,g_n$ are bounded and that the sequence $\alpha_n$ is convergent to $\alpha.$

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