section_path: Types of limits/In sequences/Real numbers
section_name: Real numbers
paragraph_text: Real numbers
  
  The expression 0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999,... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1.
  
  Formally, suppose a, a,... is a sequence of real numbers. When the limit of the sequence exists, the real number L is the limit of this sequence if and only if for every real number ε > 0, there exists a natural number N such that for all n > N, we have | a − L | < ε. The common notation lim n → ∞ a n = L {\displaystyle \lim _{n\to \infty }a_{n}=L} is read as:
  
  The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value | a − L | is the distance between a and L.
  
  Not every sequence has a limit. A sequence with a limit is called convergent; otherwise it is called divergent. One can show that a convergent sequence has only one limit.
  
  The limit of a sequence and the limit of a function are closely related. On one hand, the limit as n approaches infinity of a sequence { a } is simply the limit at infinity of a function a (n) —defined on the natural numbers { n }. On the other hand, if X is the domain of a function f (x) and if the limit as n approaches infinity of f (x) is L for every arbitrary sequence of points { x } in X − x which converges to x, then the limit of the function f (x) as x approaches x is equal to L. One such sequence would be { x + 1/ n }.