Originally Posted by

**deinol** When an assumption is given to you at your progress (up to some page of a beginners' Analysis book, some point of your course etc.)

For instance, was the exponential function properly defined? Was it proven to be strictly increasing and continuous? Was even the continuity of a function defined? When the answers to these are given to us, we would be able provide you with what you've requested.

I doubt this.

Firstly, you will have to deal with what you mean by "if $\displaystyle t<x$", though it is a minor problem, compared to this: even if $\displaystyle t<x \Leftrightarrow b^t \in B(x)$, we still have $\displaystyle sup(B(x))=b^x$. Think of x being irrational, then the equality part of the inequality sign would mean nothing.

One more note is, even if you were right above, you still haven't actually proven $\displaystyle b^x$ is the least upper bound.

You might have thought "since the supremum is a fine line to cross, and proving we have crossed the fine line, by using slightly smaller numbers to generate B(x), we can somehow prove $\displaystyle b^x$ is the supremum". Sometimes this is true (not in this case), but still you may have to elaborate.

This are some general tips for deciding what you should assume:

1. Stick to the proven theorems given to you, unless it is something trivial (e.g. various basic properties of the natural numbers).

2. Check where did you theorems come from - do not try to prove a premise in a theorem using the theorem.

>Opalg

I think defining $\displaystyle e^x$ as a power series is enough to omit "..define b^x as the limit of b^t...". And the continuity of $\displaystyle e^x$ already implies a proper definition of b^x when x is anyway real.

(Edited many times due to my careless and impatient proof-reading... this will be the last 02:18 16/11/2008)