Hmh, some educated but not self-employed by the gist of it, amusing themselves.

Individual- You're right of course. Idle hands.

Sara242-
There's the Boundedness Theorem that seems related here- though I'm not so good at "Pure Maths".

Try this link
http://socratic.org/precalculus/functions-defined-and-notation/boundedness

Or this one
http://math.stackexchange.com/questions/1365882/boundedness-theorem-for-continuous-functions-question

Some very good answers being posted though on this thread

Think about the base you are using. 0.11111... in base 9 is 0.1.

"Logic proves: All opinions -for and against are equally valid."

Illogical.

"Logic proves: All opinions -for and against are equally valid."

Logic would be that some opinions are valid & some not.

Aidan: You said in your first post, "Mathematics is totally consistent. ".
This is not a provable! (This is what is known as Godel's First Incompleteness Theorem.) So it may well be true, but we will never be able to prove it. In other words, you are stating it as a matter of faith not as a fact of maths.

Getting back to sara242's statement and where it actually fails to prove an inconsistency:
sara242 first "proof" states
1) fact is 1 is an integer 0.999.. is a non-integer
2) when an integer(1) = a non-integer(0.999...) we have a contradiction in maths"
- believe it or not both these statements are perfectly true statements about maths!
However, within standard maths it is NEVER the case that the integer(1) equals the non-integer (ie real) 0.9999 since you cannot compare integers to reals. There is no such equality operator- we only have equality operators that compare natural numbers to other naturals, integers to integers and so on- ie: you can only compare numbers of the same type. So even though (2) is a logically valid implication it is not logically sound (since the premise is never true) so we cannot draw the conclusion. That is we cannot state the conclusion by itself as a true statement.
(But note we can still state the overall (2) statement as true. Another example of this sort of thing is the statement: if squares are circles then 2+2=5. This statement is perfectly true, however squares as far as we know are not circles so we can't use this implication to claim anything about 2+2 equaling 5.)