# Thread: Convergence and Continuity of function sequences

1. ## Convergence and Continuity of function sequences

Hi,

I would like to know how to prove the following:

If fn converge pointwise to F on D and each fn is monotone increasing on D, then F is monotone increasing on D (and also for monotone decreasing).

( Do we have to make like a gn(x) = fn(x) - F(x) ...)

2. Assume F isn't increasing...then you have x<y with F(x)>F(y). Pick an epsilon smaller than half of |F(x)-F(y)|, and find an n such that f_n is within epsilon of F at both x and y.

3. But I think that we have to somehow use the fact that fn converge pointwise to F on D . . .

4. Originally Posted by zxcv
But I think that we have to somehow use the fact that fn converge pointwise to F on D . . .
That's why n exists.

5. Can you please further elaborate as it is not too clear to me.

6. Originally Posted by zxcv
Can you please further elaborate as it is not too clear to me.
Since $\{f_n\}$ converges pointwise on D, if we have $x,y\in D$ then there's some $N_x$ such that $|f_n(x)-F(x)|<\varepsilon$ for every $n>N_x$, and likewise there exists an $N_y$ for the point y. Take the maximum of those two, and continue the proof.