Proposition: Calculations with Uniformly Convergent Functions

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

Then the following functions are also uniformly convergent:

If, in addition to the uniform convergence, all $f_n,g_n$ are bounded, then

Proofs: 1

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  1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition