Analysis Rocks these days

Ah, I am so happy now :). Today I was able to understand lot of things including the proof of "Intermediate Value Theorem", "Extreme Value Theorem", "Interior Extremum Theorem", "Rolle's Theorem", "Mean Value Theorem". The key to understanding all these was the good amount of time I spent understanding the "Continuity", "Limit proofs" and the relation to Compactness and its theorems. So there are 4 equivalent ways to state the same thing that a function has a limit at any pt c, or a function is continuous at a pt c. The first one is the definition, the second one is delta epsilon, the third one is topological definition and the fourth one is sequence definition. Also these two proofs makes the life much simple.
1. On a compact set a continuous function has a compact image.
2. On a connected set a continuous function has a connected image.
The Proof of Extreme Value Theorem follows from the compactness.
The Extreme Value theorem says that a continuous function on a compact domain has a maximum and minimum. Since the domain of the function is compact it implies the range is also compact. Thus the range is closed and bounded. Therefore it has a max and a min.
The Rolle's theorem proof is just the application of Interior Extremum Theorem which says that if a function is differentiable in the open interval then it has a max and a min and the derivative is zero there.
The proof of Mean Value theorem follows if we subtract the equation of the line from the function and differentiate.
The proof of Intermediate Value theorem was done by choosing an epsilon such that it gives a contradiction for continuity. (LOL) I am not sure anybody reading this last line will understand this, what I mean by that. But if you been studying this proof for sometime. It will definitely makes sense. Otherwise check the proof on Wikipedia. Its pretty good.
Finally if you are wondering about what is written on the white board at my desk, well its a proof of if domain is compact then range is also compact !