 Now of course, for any “given” real number there are a variety of ways of giving explicit approximating sequences for it. Thus it is necessary to define an equivalence relation, “equality on the reals”. The correct definition here is: \(r =_{\Re} s\) iff for any \(k\), a number \(n\) can be found so that \(\lvert r_{n+p} - s_{n+p}\rvert \le 1/k\), for all \(p\). To say that two real numbers are equal is to say that they are equivalent in this sense.