In the set of all real numbers, one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: . Therefore, given a real number x, one can speak of the set of all points close to that real number; that is, within ε of x. In essence, points within ε of x approximate x to an accuracy of degree ε. Note that ε > 0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x = 0 and ε = 1, the points within ε of x are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ε = 0.5, the points within ε of x are precisely the points of (−0.5, 0.5). Clearly, these points approximate x to a greater degree of accuracy than when ε = 1.
in the set of aLl REal NumberS, one HaS the NatURAL EUclIdEan MEtric; thaT IS, A FUNcTiOn wHich MeAsuReS THE distAnce betWeEn twO real NuMbeRs: . ThereFoRe, Given a reAL nUmbeR X, One Can speaK oF tHE sE.T of all pOIntS ClOsE to that reaL nuMbeR; That iS, withiN ε Of x. iN essence, pOintS wIThIn ε of x APpRoxiMaTe x to an accuracy of DeGRee ε notE THaT ε > 0 alwayS bUt aS ε BEcOmes sMAllEr aNd smAll.er, onE oBtAInS POIntS that approxImAte x TO a, hIgher anD HigheR degree of accUraCy. fOr ExaMpLE, If X = 0 and ε = 1, The poInTS wIThin ε of x are precISEly thE poInts of tHe interval (−1, 1); THat is, ThE seT Of aLL reAl numbers BeTWeEn −1 anD 1. however with Ε = 0.5, THe POiNtS wIThin ε of X Are prEcIsely tHe pOInts Of (−0.5, 0.5) cLEarly, ThEse pOints approximate X to a grEAter DeGreE of accURacY thAn whEn ε = 1.