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Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle. Most of the time a mathematical statement is classified with one the words listed above. However, I can't seem to find definitions of them all online, so I will request your aid in describe/define them.
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The important thing about a convergent sequence is that the convergent behavior has nothing to do with the first few terms; it doesn't have anything to do with the first hundred terms, or the first billion terms, or any given number of terms.
In other words, your teacher's definition does not say that a sequence is bounded if every bound is positive, but if it has a positive bound. The sequence $(0,0,\ldots)$ has indeed a positive bound: $1$, for example (in fact, every positive real number is a bound for this sequence!)
Question 2: Here are four examples from my bookshelves:. Derek Robinson's A Course in the Theory of Groups, 2nd Edition (Springer, GTM 80), defines a group as a semigroup (nonempty set with an associative binary operation) that has a right identity and right inverses (page 1; he proves they also work on the left in 1.1.2, on page 2).
Hopefully, this sheds some light on why the formal definition is taken to be exactly the way it is, in order to match the intuition. Finally, we come to question 3: what does the backwards definition actually describe? The answer: every function.
A definition introduces new expressions to your language. That is, if the terms "quadrilateral" and "right angle" are known, you can introduce the new notion of "rectangle" with a definition as above. A theorem on the other hand involves known (defined) notions and expresses a provable statement about these.
The first definition is equivalent to this one (because for this limit to exist, the two limits from left and right should exist and should be equal). But I would say stick to this definition for now as it's simpler for beginners. The second definition is not rigorous, it is quite sloppy to say the least.
The difficulty with this definition is that you need to have a guess for what the integral actually is before proving that the function is integrable. This is why upper and lower sums are actually so important, and why most definitions require them. It can be proven that Darboux integrability implies Riemann-integrability.
My best description is that "inductive definition" is more common when we are defining a set of objects "out of nothing", while "recursive definition" is more common when we are defining a function on an already-existing collection of objects. A prototypical inductive definition is the following definition of the set of natural numbers: