I need to understand as to why the following series fn(x) = x/(1+n*x^2) is point wise convergent (as mentioned in the book of Ross) and not [obviously] uniformly convergent.
The relevant equation used is that lim (n-> infinity) sup|(fn(x) - f(x))|= 0 implies uniform convergent.---- (1)
The Attempt at a Solution
It is obvious lim (n-> infinity) fn(x) = f(x) = 0 for x not equal to zero. And when x=0, fn(0) = 0 and hence as n-> infinity, fn(0) = f(0) =0.
As mentioned in Ross and ( I can see) that fn(x) is pointwise convergent. But, it looks like for all x, the function fn(x) converges to f(x)=0. So, I am unclear as to why it is not uniform convergent? Please clarify.
The book further uses the theorem (1) to prove uniform converfence, which I can understand.