For n 1, 2, brvbar , let F n and F be d f s such that And let F be continuous Then show that Uniformly in x Atilde Atilde For Icirc micro 0, choose a and b sufficiently small and sufficiently large, respectively, so that F ( a ) Icirc micro 3, F ( yen ) F ( b ) Icirc micro 3 Next, partition a, b by...

For 0 select a b so that Next F being continuous is uniformly continuous in a b Then ...

For n = 1, 2,¦, let F n and F be d.f.s such that And let F

Question:

Forn= 1, 2,€¦, letF_nandFbe d.f.s such that

And let F be continuous. Then show that

Uniformly in x ÃŽ Ã‚.

For Îµ > 0, choose a and b sufficiently small and sufficiently large, respectively, so that F(a) < Îµ/3, F (¥) €“ F(b) < Îµ/3. Next, partition [a, b] by a = x₀ < x1 < €¦ < xk €“ 1 < xk = b, so that F(xj) €“ F(xj €“ 1) < Îµ/3, j = 1, €¦., k. Finally, by taking x ÃŽ [a, b], or x < a, or x > b, show that |F_n(x) €“ F(x)| < Îµ for all x, provided n ³ N some integer independent of x ÃŽ Ã‚?