Prove: {Fn} converges to F on [a,b] and Fn is nondecreasing for all n; show F is too

Homework Statement

Prove: If {Fn} converges to F on [a,b] and Fn is nondecreasing for each n$$\in$$ N, then F is nondecreasing.

n/a

The Attempt at a Solution

First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove uniformly and then it holds for pointwise as well?

My work so far:

Suppose Fn converges to F and Fn is nondecreasing: i.e., for all n $$\in$$ N, x1 $$\leq$$ x2 $$\Rightarrow$$ Fn(x1) $$\leq$$ Fn(x2).

So $$\forall$$ $$\epsilon$$ > 0, $$\exists$$ N1 $$\in$$N, such that $$\forall$$ n $$\geq$$ N1, sup{|Fn(x)-F(x)|: x $$\in$$ [a,b]} < $$\epsilon$$.

Equivalently, ||Fn(x) - F(x)||$$_{}[a,b]$$ < $$\epsilon$$

| ||Fn(x)||$$_{}[a,b]$$ - ||F(x)||$$_{}[a,b]$$ | < $$\epsilon$$ by the reverse triang. inequal

||F(x)||$$_{}[a,b]$$ - $$\epsilon$$ < ||Fn(x)||$$_{}[a,b]$$ < ||F(x)||$$_{}[a,b]$$ + $$\epsilon$$

LeonhardEuler
Gold Member

First, it doesn't say if Fn converges pointwise or uniformly, so I'm not entirely sure how to deal with that. Just prove uniformly and then it holds for pointwise as well?

Since a sequence of functions that converges uniformly also converges pointwise, you actually want to prove it for the case of pointwise convergence (A weaker hypothesis gives a more general theorem). It will also be much more direct to prove this way since you will be comparing two points and taking limits at those points.

Dick