What is the rule of conditional proof and how it is used in a proof (particularly in an Analysis proof)?
Give an example?
See Wikipedia (skip all fancy words and get the idea; see also the sample derivation). Google returns many other results.
The vast majority of theorems in mathematics have the form of implication, and they have conditional proofs. Exceptions are absolute statements like "$\displaystyle \pi$ is transcendental" and its corollary "Quadrature of the circle is impossible". (To dig deeper, even in such statements implication often hides under a definition.)
In analysis one has theorems like "If a function is continuous on a segment with endpoints, then it is bounded". One assumes that an arbitrary function f is continuous, on this basis one proves that it is bounded. At this point it does not mean that an arbitrary function is bounded. One needs to "close" the open assumption that f is continuous. The result is a statement in the form of implication.
Further details depend on your course: what the context is, how deep you study this concept (logic for philosophers vs. discrete math for computer scientists) and how formal the proofs are that you consider.
Thanks .
For example in the case of group theory where the statement :
the identity element is unique
IS not of the implication type ,do we use the rule of the conditional proof ?
Also in Analysis where we are asked to prove,for example:
$\displaystyle \lim_{n\to\infty}x_{n} =x$ do we use the rule of conditional proof?
If yes ,how?
First I think you are letting concerns of formal logic get in your way of doing mathematical proofs.
Second, I would argue that the theorem In a group the identity is unique. is indeed a form of implication.
In that it is an A proposition: All groups have a unique identity.
$\displaystyle \text{If }G\text{ is a group, then }G\text{ has a unique identity.}$