Question

what is the difference between the pointwise convergent sequence and uniform convergent sequence??​

Answer 1

Answer:

Look at this as a game between two people (practically, this can't happen but let our imagination flow). Each game actually is a "translation" of the corresponding definition.

You want to prove pointwise convergence. Lets make a game for it.

Step 1. You chose some x from the domain.

Step 2. The opponent chooses some ϵ>0.

Step 3. You try to find an N∈N such, that ∀n≥N, |fn(x)−f(x)|<ϵ.

Additional Rules: Steps 2 and 3 must repeat until your opponent is convinced that for the particular x you picked up, whatever ϵ he tells you, you'll be able to find such an N. You will be able to select another x from the domain, only after you have convinced your opponent. The games is over when you have done this for all x in the domain.

As you may have already noticed, there is a precise correspondence between the definition and the game. Step 1 corresponds to binding the variable x. When x takes a value, this value becomes fixed, so that we can range over ϵ, which corresponds to Step 2. After a value is chosen for ϵ, this value remains fixed. In Step 3 you find the proper N within the context of already chosen values for x and ϵ. This is why in pointwise convergence N depends on both x and ϵ.

Now, you want to prove uniform convergence. The game here changes.

Step 1. Your opponent chooses an ϵ>0.

Step 2. You try to find an N such that ∀n≥N, |fn(x)−f(x)|<ϵ,∀x.

The difference here is that you don't check each x separately. On the contrary, your opponent gives you an ϵ and the N you have to find refers to all x in the domain, not just one you picked up. It is like considering the range of x all at once. The game ends when your opponent is convinced that you'll find such an N, whatever ϵ he tells you

Step-by-step explanation:

hope this will help u