00:03
Well, for a, well, suppose fn uniformly converges to f and gn uniformly converges to g.
00:24
Then we want to show what we claim is that fn plus gn uniformly converges to f plus g.
00:36
And so let's show this well what does it mean to uniformly can we need to look at soup of x in e of uh f n plus g n minus f minus g right well by triangle inequality this is the same thing as soup of x in e of, because i can split this up as fn minus f plus gn minus g.
01:29
Okay, but then i can take the soup of a sum must be less than equal to the sum of the soups.
01:38
So i can make the soup separately.
01:42
So i can take soup of x and e of fn of fn.
01:48
Minus f plus the soup of x in e of gn minus g and g and but you know that fn converges uniformly and gn converts uniformly so that tells you that these two go to zero right because this is just the definition of uniform convergence and since this is bigger than equal to zero this must mean by the squeeze theorem that this thing also goes to zero and so you have uniform convergence okay now for b let's look at the product so in particular we're gonna look at soup of x in b of f n g and minus fg okay now we're gonna write this in a different way, since we want to use triangle inequality again.
03:02
And the way we're going to write this is, i'm going to write this as fngn minus fgn plus fgn and then do minus.
03:21
Okay, so all i've done is i've added and subtracted the same value, which i'm allowed to do.
03:26
And now i'm going to split this up, again, using the same type of inequality as above.
03:32
So i'm going to split this up as soup of x and e of, well, note that i'm going to split this up in this way.
03:41
So i'm going to split this up and then this up.
03:46
And note that the gn just comes out.
03:49
So you have gn and then you have fn minus f.
03:55
And then again, i'm going to split the soup as well, f and then i have gn minus.
04:05
G...