Necessary and sufficient

In mathematics, the phrase that some condition is "necessary and sufficient" for some other statement is frequently used, in particular, in the statement of theorems, when justifying a step in a proof, or to introduce an alternative version for a definition.

In mathematics, the phrase "necessary and sufficient" is frequently used, for instance, in the formulation of theorems, in the text of proofs when a step has to be justified, or when an alternative version for a definition is given.

To say that a statement is "necessary and sufficient" to another statement means that the statements are either both true or both false.

Another phrase with the same meaning is "if and only if"

A statement A is (a) necessary and sufficient (condition) A statement A is necessary and sufficient

                 is a necessary and sufficient condition

for another statement B if it is both a necessary condition and a sufficient condition for B, i.e., if the following two propositions both are true:

• A is (a) necessary (condition) for B,

The statement

• A is a necessary condition for B,

       (or shorter: is necessary for) B,

means precisely the same as each of the following statements:

• B is false whenever A does not hold, or, equivalently.
• B implies A.
• If A is false then B cannot be true

• A is (a) sufficient (condition) for B,
• A is a sufficient condition for

       (or shorter: is sufficient for) B,

means precisely the same as each of the following statements:

• B holds whenever A is true.
• B holds whenever A is true.
• A implies B.

Example

For a sequence of positive real numbers to converge against a limit

• it is necessary that the sequence is bounded
• it is sufficient that the sequence is monotone decreasing
• it is necessary and sufficient that it is a Cauchy sequence.

A sequence $(a_{n}),\ 0\leq a_{n}\in textrmR$ $\lim _{n\to \infty }a_{n}=a\Rightarrow a_{n}\ {\text{bounded}}$ $\lim _{n\to \infty }a_{n}=a\Leftarrow a_{n}\ {\text{monotone decreasing}}$ $\lim _{n\to \infty }a_{n}=a\Leftrightarrow a_{n}\ {\text{is a Cauchy sequence}}$

$\lim _{n\to \infty }a_{n}=a\Rightarrow a_{n}<C\in {\textrm {R}}$ $\lim _{n\to \infty }a_{n}=a\Leftarrow (\exists C)(\forall n)a_{n}<C\in {\textrm {R}}$ $\lim _{n\to \infty }a_{n}=a\Leftarrow (\forall \epsilon >0)(\exists N\in {\textrm {N}})$