Am 04.10.2013 11:50, schrieb quasi: > Roland Franzius wrote: >> >> Take any sequence of rational approximations of pi >> >> r_i = m_k/n_k -> pi eg the continued fraction for atan 1. >> >> Since pi is not rational, there is no upper limit on n_k. >> >> The continued fractions are the best rational approximations, >> the limit is never more than 1/n_k away. >> >> Take a subsequence with denominator n_(k_j) of more than >> exponential growth eg e^(n_k_j^2). >> >> Then >> >> abs(sin n_(k_j)) >> = abs(sin ( m_(k_j) pi+ eps_j)) >> = abs(sin eps_j) >> < 1/2 eps_j > > How do you justify the claim > > abs(sin(eps_j)) < (1/2)*eps_j > > ?? >
We called it the epsilon trick.
In full generality, in a logic oriented freshmans analyis course it was once shown, that you can change epsilon by any positive factor if you find in the course of a proof that the outcome is bigger than epsilon.
The three epsilon trick is to show that something is less than epsilon/3, so there is some space left for an unforseen blow up by a triangle inequality.