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

", though it is a minor problem, compared to this: even if

, we still have

. 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

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

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

as a power series is enough to omit "..define b^x as the limit of b^t...". And the continuity of

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)