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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.
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.