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k3k3

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**Showing the sum of functions are uniformly continuous**

## Homework Statement

Suppose f and g are uniformly continuous on an interval I. Prove f + g are uniformly continuous on I.

## Homework Equations

## The Attempt at a Solution

Let ε >0

By definition, since f and g are uniformly continous on I, there exists a [itex]\delta[/itex]_1 such that |f(y)-f(x)| < ε for all x,y in I that satisfy |x-y| < [itex]\delta[/itex]_1

Similarly, for g, there exists a [itex]\delta[/itex]_2 such that |g(y)-g(x)| < ε for all |y-x| < [itex]\delta[/itex]_2

Then, for all x,y in I, |f(y)+g(y)-(f(x)+g(x))| ≤ |f(y)-f(x)|+|g(y)-g(x)| by the triangle inequality. This implies |f(y)+g(y)-(f(x)+g(x))| < 2ε

Choose [itex]\delta[/itex]=min{[itex]\delta[/itex]_1,[itex]\delta[/itex]_2} * 1/2

Then |f(y)+g(y)-(f(x)+g(x))| < 2ε for all x,y in I that satisfy |y-x| < [itex]\delta[/itex]

∴f+g is uniformly continuous on I.

Is this correct?

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