Let X R have the topology I U C X 1 epsilon U or U Phi Le

Let X = R have the topology I = {U C X: 1 epsilon U or U = Phi}. Let s be the sequence in X given by s_n = 1 for each n epsilon Z^+. Does s converge? If it does, find all numbers to which it converges.

Solution

The given sequence is the constant sequence ( 1, ,1, 1, ... )

A sequence in a topological space X converges if all the terms of the sequence are in one of the open sets of the given topology after a certain stage. All the terms of this sequence is 1 which is in some of the open set ( all except the null set ) of the given topology. hence the given sequence is convergent to 1 in X