Sequence of more than one index
Sequence of more than one index [ edit ] Sometimes one may also consider a sequence with more than one index, for example, a double sequence ( � � , � ) . This sequence has a limit � if it becomes closer and closer to � when both n and m becomes very large. Example [ edit ] If � � , � = � for constant c , then � � , � → � . If � � , � = 1 � + � , then � � , � → 0 . If � � , � = � � + � , then the limit does not exist. Depending on the relative "growing speed" of n and m , this sequence can get closer to any value between 0 and 1. Definition [ edit ] We call � the double limit of the sequence ( � � , � ) , written � � , � → � , or lim � → ∞ � → ∞ � � , � = � , if the following condition holds: For each real number � > 0 , there exists a natural number � such that, for every pair of natural numbers � , � ≥ � , we have | � � , � − � | < � . [10] In other words, for every measure of closeness � , the sequence's terms are eventually