Many proofs in transcendental number theory reach a contradiction by coming up with a quantity which on the one hand is a positive integer, and on the other hand is less than one.

A particular simple example is the proof that $e$ is irrational. Suppose $e = p/q$. Then for $r \geq q$,
$$r!e = \sum_{n \leq r} \frac{r!}{n!} + \frac{1}{r+1} \left(1 + \frac{1}{r+2} + \frac{1}{(r+2)(r+3)} + \cdots \right).$$
Now consider the quantity
$$ \frac{1}{r+1} \left(1 + \frac{1}{r+2} + \frac{1}{(r+2)(r+3)} + \cdots \right). $$
On the one hand, it's a positive integer. On the other, as $r\to\infty$ it tends to 0.

In other proofs we might come up with (the analog of) some explicit $r$ for which the quantity is less than $1$, rather than showing that it tends to $0$.

Integers (strictly) between 0 and 1 form the basis of transcendental number theory?
Currently in Khojand, Tajikistan and wanting to take a taxi or something tomorrow to Dushanbe.  The taxi drivers here are like vultures - I was surrounded by the horde of them at the station outside town where the border taxi guy dropped me off.  I've been quoted US$40 for a ride to Dushanbe tomorrow, in a shared cab, or $100 for just me (I'd go shared) but it's in a regular car and have no way of knowing if that's good or not.  Googled without much luck.  I've been told you should only take a jeep, or 4wd, but everyone says different things (taxi drivers) and it's hard to know who to trust!

How it turned out - see answer at bottom, successfully made it to Dushanbe without being ripped off too badly ;)
