Science Quiz / Mathematical Definitions

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Can you name the given properties?

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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.The sequence (a_n) ... to ..
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.(a_n) is a ... sequence.
There exists a real M such that |a_n| less than or equal to M for all n.The sequence (a_n) is...
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.x is a(n) ... of (a_n).
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.(b_n) is a ... of (a_n).

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