Prove the sequence converges uniformly

[wackikat]

Homework Statement

Let f_n be a sequence of function whcih converges pointwise on [0,1] where each one is Lipschitz with the same constant C. Prove that the sequence converges uniformly.

Homework Equations

A function is called Lipschitz with Lipschitz constant C if |f(x)-f(y)| <= C|x-y| for all x,y in its domain.

Let f_n be a sequence of functions defined on a set S. f is the pointwise limit of f_n if for all t in S lim n to infinity f_n(t) = f(t)

The Attempt at a Solution

I know that somehow I must show if for all epsilon > 0 there exists N in naturals such that sup|f_n(t) -f(t)| < epsilon if n>N for all t in [0,1]
Or show the Uniform Cauchy Criterion holds : for all epsilon > 0 there exists N in naturals such that |f_n(t) -f_m(t)| < epsilon for all m,n > t for all t in [0,1].

 

[Focus]

Go for the uniform Cauchy, use the triangle inequality and the fact that |x-y| could at most be 1.

 

[yyat]

Hint: Pointwise convergence implies uniform convergence on any finite set of points. Since [0,1] is compact, you can choose points x₁,...,x_k such that the distances between consecutive points is arbitrarily small.

 

[wackikat]

I've tried using Cauchy, but I just seem to end back with a term I started with.
Here's what I tried.
|f_n(x) -f_m(x)| = |f_n(x) + f_n(y) + f_n(y) - f_m(x)| <= |f_n(x) + f_n(y)| + |f_n(y) - f_m(x)| <= C|x-y| + |f_n(y) - f_m(x)| = C|x-y| + |f_n(y) -f_n(x) + f_n(x) - f_m(x)| <=
2C|x-y| + |f_n(x) - f_m(x)|

As for yyat's hint, I don't believe that is true. The sequence of funtions could converge to a discontinuouse f(x) which would mean there could not be uniform convergence.
If we knew Pointwise convergence implies uniform convergence on any finite set of points then we would not need the fact that the functions are Lipschitz.

 

[yyat]

As for yyat's hint, I don't believe that is true. The sequence of funtions could converge to a discontinuouse f(x) which would mean there could not be uniform convergence.

Any function defined on a finite set of points is continuous.

If we knew Pointwise convergence implies uniform convergence on any finite set of points then we would not need the fact that the functions are Lipschitz.

Why? You want to prove uniform convergence on [0,1], which is not a finite set. The Lipschitz continuity is crucial here.