The idea of compactness, which is a property of some spaces, is intuitively based on the degree of fullness of some set equipped with a metric, (X,d). The technique of measuring that fullness is convergence. Since any set for which compactness is to be assessed must be bordered, it must also contain enough points within that set such that there exists some sequence in the set that converges at some given point in the sequence indexed by a value k. The concept of sequence convergence is so fundamental to analysis in the same way the delta-epsilon is so important to the definition of limits.
The idea of compactness, which is a property of some spaces, is intuitively based on the degree of fullness of some set equipped with a metric, (X,d). The technique of measuring that fullness is convergence. Since any set for which compactness is to be assessed must be bordered, it must also contain enough points within that set such that there exists some sequence in the set that converges at some given point in the sequence indexed by a value k. The concept of sequence convergence is so fundamental to analysis in the same way the delta-epsilon is so important to the definition of limits.