Hi,
is the following sequence of functions uniformly convergent for ?
whereas
is a constant
are constants
are constants
I think the pointwise limit is given by
Is this correct?
Does also converge uniformly to ?
Can anybody help me?
Thanks!
Hi,
is the following sequence of functions uniformly convergent for ?
whereas
is a constant
are constants
are constants
I think the pointwise limit is given by
Is this correct?
Does also converge uniformly to ?
Can anybody help me?
Thanks!
I was hoping you wouldn't ask that, because it doesn't seem to be obvious. Here's one approach that appears to work.
First, the functions increase monotonically to as , and it follows from Dini's theorem that the convergence is uniform on any finite interval [0,T] (where T is to be chosen in the next paragraph). Thus given there exists with such that for all whenever
Next, the function is continuous and hence bounded when and , say Hence whenever Since g(a,h) is close to g(a,0) for h sufficiently small, that should be enough to show that converges to uniformly on [0,1].