Uniform continuity of composite function

[estro]

I'll be very thankful is someone will tell me where I'm wrong.

We know:
1) f is uniform continuous.
2) g is uniform continuous.

We want to prove:
fg(x) is uniform continuous.

proof:
from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e
from 2 we know -> for every |x-y|<d exists |g(x)-g(y)|<d_0
let a=g(x) and b=g(y) then
for every x,y |x-y|<d exists |fg(x)-fg(y)|<e.

Sorry for my poor formulation, English is not my mother tongue.

 

[zhentil]

What's wrong with it?

 

[estro]

I'll be more then happy if this is right...
I'm just not sure.

 

[zhentil]

Yes, it's correct.

 

[Bacle]

Estro:

You need to be careful to specify where (in their respective domains) f and g
are uniformly continuous. Otherwise your statement is not true. I think Zhentil
assumed f,g were everywhere unif. continuous.

 

[estro]

Oh, thanks for the remark.
I indeed intended for every x,y in R.