Question

show that if f and g are uniformly continous on A in R and if they are both bounded on A then their product fg is uniformly continuous show that if f and g are uniformly continous on A in R and if they are both bounded on A then their product fg is uniformly continuous @Mathematics

Answer 1

start with
\[|f(x)|<M_1, |g(y)|<M_2\] for all x,y in A

Answer 2

then consider
\[|f(x)g(x)-f(y)g(y)|=|(f(x)-f(y))g(x) + f(y)g(x) - f(y)g(x)|\]
\[\leq | g(x)(f(x)-f(y)) | + | f(y)(g(x)-g(y)) |\]
\[\leq |g(x)||f(x)-f(y)|+|f(y)||g(x)-g(y)|\] is the standard trick i think for this one. boundedness gives you what you need for the first part, and uniform continuity gives the rest.

Answer 3

can i assume that f and g are both bounded by M? that is the only difference between what we have/

Answer 4

you need that they are both bounded. otherwise proof fails and it is in fact false.

Answer 5

oh yes, i understand the question. you can make M = max of bounds , works the same