I need help proving or disproving if the following function converges uniformly to its point wise limit

let fn(x)= 1 if x is in [2n-1, 3n-1]
0 if x is not in [2n-1, 3n-1]

I found the pointwise limit to be 0 but I am having trouble proving uniform convergence to the point wise limit

any help appreciated

There is no uniform convergence in this case.

If it would, you should, for every ε > 0, find some rank N, with the following property :

for every n > N, | fn(x) - f(x) | < ε, for every real x.

This is obviously wrong, since the limit is the function f(x) = 0 (ie : for every real x, fn(x) tends to 0 as n approaches + ∞).

Choose any 1 > ε > 0, choose any rank N, there is always an integer n > N and a real x such that | fn(x) - 0 | > ε (you can consider any x in the interval [2n-1 ; 3n-1]).