Proving Uniform Convergence of fn to f on [a,b]

[lola21]

Homework Statement

Suppose:
fn goes to f pointwise on [a,b]
For each c in (a,b) fn goes to f uniformly on (c,b), and for some M, fn(x) is less than M for all n, all x in [a,b]
Prove fn goes to f uniformly on [a,b]

Homework Equations

The Attempt at a Solution

I think I proved that we can choose c sufficiently close to a such that fn converges to f uniformly on (a,b). I can't see how to show that fn converges uniformly when we include the endpoints. Is more information necessary? If so, what do we need to know?

 

[mighty2000]

it is important to approach problems in a systematic and logical manner. In this case, we are given some information about the convergence of a sequence of functions, but we need to prove that it converges uniformly on a particular interval. Let's break down the problem and see what we can do with the given information.

First, we know that fn goes to f pointwise on [a,b]. This means that for any x in [a,b], the sequence of values fn(x) approaches the value f(x). In other words, as n gets larger, the values of fn(x) get closer and closer to f(x).

Next, we are given that for each c in (a,b), fn goes to f uniformly on (c,b). This means that for any c in (a,b), we can choose a value of n such that for all x in (c,b), the difference between fn(x) and f(x) is less than some small positive number ε. In other words, we can make the values of fn(x) arbitrarily close to f(x) for all x in (c,b) by choosing a large enough n.

Finally, we are told that fn(x) is less than M for all n and all x in [a,b]. This means that the sequence of functions is bounded on the interval [a,b], as all values of fn(x) are less than some fixed value M.

Now, let's think about how we can use this information to prove that fn converges uniformly on [a,b]. Since we know that fn goes to f pointwise on [a,b], we can say that for any x in [a,b], there exists some value of n such that for all n greater than this value, fn(x) is within ε of f(x). This is true for all x in [a,b], so we can say that for all x in [a,b], there exists some value of n such that for all n greater than this value, |fn(x) - f(x)| < ε. This is the definition of uniform convergence, so we have shown that fn converges uniformly on [a,b].

In summary, by using the given information about pointwise and uniform convergence, as well as the boundedness of the sequence of functions, we have been able to prove that fn converges uniformly on [a,b]. This is a key result in analysis and can be applied to many different problems in