Processing math: 100%

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


Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs


References

Bibliography

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