In this installment, we introduce limits and use them to lead to the notion of complete sets. This is the last piece we need to define real numbers next month.

I don’t know if you have used the Archimedean axiom for the reals: For any two positive reals x and y, there is an integer n such that n*x > y. This, combined with Bernoulli’s inequality ((1+x)^n >= 1 + nx) will allow you to show that lim_{n -> infinity} r^n = 0 for 0 < r < 1.

I can provide more info if you want. email me at mjcohen@acm.org.

mjcohen

May 04, 2013@ 22:13:50I don’t know if you have used the Archimedean axiom for the reals: For any two positive reals x and y, there is an integer n such that n*x > y. This, combined with Bernoulli’s inequality ((1+x)^n >= 1 + nx) will allow you to show that lim_{n -> infinity} r^n = 0 for 0 < r < 1.

I can provide more info if you want. email me at mjcohen@acm.org.

W. Blaine Dowler

May 05, 2013@ 04:57:38I’ve got it in my resources, but it isn’t strictly necessary to define the real number set so I haven’t used it yet. It’ll come up later.