Difference between a lemma, proposition, theorem or corollary

(Reference: Analysis I, Third Edition, Prof Terence Tao, Hindustan Book Agency)

From a logical point of view, there is no difference between a lemma, proposition, theorem or corollary — they are all claims waiting to be proved. However, we use these terms to suggest different levels of importance and difficulty. A lemma is an easily proved claim which is helpful for proving other propositions and theorems, but is usually not particularly interesting in its own right. A proposition is a statement which is interesting in its own right, while a theorem is a more important statement than a proposition which says something definitive on the subject, and often takes more effort to prove than a proposition or lemma. A corollary is a quick consequence of a proposition or theorem that was proven recently.

Many thanks to Prof. Terence Tao !

From Nalin Pithwa !

Interesting questions suggested by Prof. Terence Tao

Reference Analysis I by Terence Tao, Third Edition, Hindustan Book Agency.

1. What is a real number ? Is there a largest real number? After Zero, what is the “next” real number (that is, what is the smallest positive real number)? Can you cut a real number into pieces infinitely many times ? Why does a number such as 2 have a square root, while a number such as -2 does not? If there are infinitely many reals and infinitely many rationals, how come there are “more” real numbers than rational numbers?
2. How to take the limit of a sequence of real numbers ? Which sequences have limits and which have don’t ? If you can stop a sequence from escaping to infinity, does this mean that it must eventually settle down and converge ? Can you add infinitely many real numbers together and converge ? Can you add infinitely many real numbers together and still get a finite real number ? Can you add infinitely many rational numbers together and end up with a non-rational number ? If you rearrange the elements of an infinite sum, is the sum still the same ?
3. What is a function ? What does it mean for a function o be continuous ? differentiable ? integrable ? bounded ? Can you add infinitely many functions together ? What about taking limits of sequences of functions ? What about taking limits of sequences of functions ? Can you differentiate an infinite series of functions ? What about integrating ? If a function takes the value 3 when x=0 and 5 when x=1 (that is, and ), does it have to take every intermediate value between 3 and 5 when x goes between Zero and One ? Why?

Further Prof. Tao remarks: You may already know how to answer some of these questions from your calculus classes, but most likely these sorts of issues were only of secondary importance to these courses; the emphasis was on getting you to perform computations, such as computing the integral of from x=0 to x=1. But now that you are comfortable with these objects and already know how to do all the computations, we will go back to the theory and try to really understand what is going on.

If you are interested further, please refer the above text.

Cheers, cheers, cheers,

Nalin Pithwa