1. ## Pointwise convergence help

ok guys i'm trying to get my head around pointwise convergence.

I'm working through an example, I = [0,1]

fn(x) = 0 ,for x=0
= ((-1)^n)n ,for x E (0,1/n]
= 0 ,for x E (1/n,1]

pick some N large enough so that it satisfies 1/N < x. Then fn(x) = 0 for all n less than or equal to N and so fn(x) ---> 0

I can't seem to see why how picking an N proves the convergence?

any help would be much appreciated

2. Originally Posted by maths52
ok guys i'm trying to get my head around pointwise convergence.

I'm working through an example, I = [0,1]

fn(x) = 0 ,for x=0
= ((-1)^n)n ,for x E (0,1/n]
= 0 ,for x E (1/n,1]

The puppy is indexed by $\displaystyle n$. thus that is what you're looking to make a statement about. everything in this problem centers around what $\displaystyle n$ you're looking at. Now by choosing an $\displaystyle N$, you're taking a step toward putting a condition on $\displaystyle n$. Without putting a condition on $\displaystyle n$, how can you prove convergence when the sequence of functions $\displaystyle f$ only have meaning in relation to $\displaystyle n$. So you restrict the statement to those functions that have $\displaystyle n \geq N$. By doing so, you guarantee that $\displaystyle x>\frac1{N}>\frac1{n}$. In those cases, $\displaystyle f_n$ converges to 0.