Infinite sequences are sequences with infinitely many terms. The expression infinite sequence are typically indexed by natural numbers and are hence forward singly infinite sequences, which have a first term and are thus are wellordered.
One may distinguish the three kinds of infinite sequences
- Forward singly infinite sequences (forward one-way infinite sequences);
- Backward singly infinite sequences (backward one-way infinite sequences);
- Doubly infinite sequences (two-way infinite sequences).
Some sequences are known to be infinite, like the sequence of prime numbers (A000040, see Euclid's proof that there are infinitely many primes), others are conjectured to be infinite, like the twin primes (A001097).