Why is the Archimedian Property used?

[Singularity]

We are currently busy in calculus in proving limits of series ans sequences exist and so on. We use epsilon arguments to prove these things. Somewhere in the proof, the Archimedian Property is used i.e. for every real number x there exists a natural number k such that k > x. I dont understand why this is needed. Can someone please shed some light on this for me? Help me to underdtand this method of proving things. It doenst seem at all logical to me.[:)]

[matt grime]

Switch it round: given any \epsilon>0 (ie 1/x) \exists n \in \mathbb{N} with 1/n < \epsilon so that the sequence 1/n tends to zero, and by comparison with this we can show lots of other things tend to zero. The proof is quite nice, I don't know if you've seen it - 1/n is decreasing and bounded below - it converges therefore to some limit x, then (1/n)^2 tends to x^2 and also, since it is a subsequence of the first sequence, it tends to x, thus x=x^2, ie x=0,1. Obvioulsy it isn't 1 (prove rigorously using the negation of the definition of convergence if you must). There is also the formulation that given any d>0, there is an integer n with nd>1