| Alternate Wording | Property | Hint |
| There exists a real L such that for all epsilon greater than 0, there exists a natural N such that | a_n - L | less than epsilon for all n greater than N. | |
| For all epsilon greater than 0, there exists a natural N such that |a_n - a_m | less than epsilon for all n and m greater than N. | |
| There exists a real M such that |a_n| less than or equal to M for all n. | |
| For every epsilon greater than 0, for every natural N there exists an n greater than or equal to N usch that |a_n - x| less than epsilon. | |
| There exists a function from the natural numbers to the natural numbers which is strictly increasing such that b_n = a_f(n) for all natural N. | |