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Uniform Convergence Proof

Prove: If fn converges to f uniformly and gn converges to g uniformly, then converges uniformly to f/g.

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The theorem as stated in your posting is WRONG!

The ratio f_n(x)/g_n(x) does NOT converge uniformly if at some x=x_0: g(x_0) = 0.

One of possible formulations to make the theorem CORRECT is the following:

There is a set X, there are sequences of functions f_n: X-->R and g_n: X-->R, both series converge uniformly, f_n-->f(x) and g_n-->g(x), and the limit functions have the following properties:
(a) both f and g are bounded, that is there are some F and G such that |f(x)| < F and |g(x)| < G for all x in X,
(b) g(x) has the property that there is some H>0 such that |g(x)| > H for all x in X.

Then f_n(x)/g_n(x) --> f(x)/g(x) uniformly.

This formulation can be modified.
--- it is possible to replace the real numbers R by complex numbers C
--- it can be stated "if and only if".
--- condition (b) can be replaced by "the set X is closed and g(x) not = 0 in X"
+ there can be other modifications

Below, I shall prove ...