Uniform convergence

hth

26
0
1. The problem statement, all variables and given/known data

Let fn(x) = 1/(nx+1) on (0,1) where x is a real number. Show this function does not converge uniformly.

2. Relevant equations

3. The attempt at a solution

I know why it is not uniformly convergent. Even though fn(x) goes to zero monotonically on the interval (0,1), it's not continuous on a compact interval. How would go about showing this formally/by example?
 
1,101
2
A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?
 

hth

26
0
A sequence of functions does not converge to the limit function f if there exists some epsilon > 0 such that for infinitely many n, |f_n(x) - f(x)| > epsilon for some x in the domain of f_n. As you mentioned, f here is 0. Now pick epsilon to be say, 1/4. For arbitrary n, can you choose x so that f_n(x) > 1/4?
Isn't that the solution to it if f_n(x) = x^(n)? How does that apply here?
 

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