Convergence

Definition

F: R ® R. F converges to b at a iff ("x)(x » a Þ F*(x) » b)

This definition is equivalent to the usual defnition of convergence,

i.e    ("x)("e > 0)($d > 0) [0 < |x - a| < d Þ |b - F(x)| < e]

i.    Suppose F converges in the ordinary way.

      Apply the method we mentioned earlier of proving properties of functions in R*.

      Note that the convergence is a statement in R and so if it holds in R it will hold in R*

      Now, if x » a, x ¹ a then 0 ¹ |x - a| < d so |b - F(x)| < e

      Since e is arbitrary, b » F*(x).

ii.   Suppose it converges according to our new definition.

      Then "e > 0 e Î R we can make the ordinary statement of convergence true in R*

      by taking d Î I.    

      So it also holds in R.

It also follows that F is continuous at a iff x » a Þ F*(x) » F*(a)