section_path: Types of limits/In sequences/Infinity as a limit
section_name: Infinity as a limit
paragraph_text: Infinity as a limit
  
  There is also a notion of having a limit "tend to infinity", rather than to a finite value L {\displaystyle L}. A sequence { a n } {\displaystyle \{a_{n}\}} is said to "tend to infinity" if, for each real number M > 0 {\displaystyle M>0}, known as the bound, there exists an integer N {\displaystyle N} such that for each n > N {\displaystyle n>N}, a n > M. {\displaystyle a_{n}>M.} That is, for every possible bound, the sequence eventually exceeds the bound. This is often written lim n → ∞ a n = ∞ {\displaystyle \lim _{n\rightarrow \infty }a_{n}=\infty } or simply a n → ∞ {\displaystyle a_{n}\rightarrow \infty }.
  
  It is possible for a sequence to be divergent, but not tend to infinity. Such sequences are called oscillatory. An example of an oscillatory sequence is a n = (− 1) n {\displaystyle a_{n}=(-1)^{n}}.
  
  There is a corresponding notion of tending to negative infinity, lim n → ∞ a n = − ∞ {\displaystyle \lim _{n\rightarrow \infty }a_{n}=-\infty }, defined by changing the inequality in the above definition to a n < M, {\displaystyle a_{n}<M,} with M < 0. {\displaystyle M<0.}
  
  A sequence { a n } {\displaystyle \{a_{n}\}} with lim n → ∞ | a n | = ∞ {\displaystyle \lim _{n\rightarrow \infty }|a_{n}|=\infty } is called unbounded, a definition equally valid for sequences in the complex numbers, or in any metric space. Sequences which do not tend to infinity are called bounded. Sequences which do not tend to positive infinity are called bounded above, while those which do not tend to negative infinity are bounded below.