Similar arguments will work for systems that are not fragments of second-order arithmetic, as long as they have $\omega$-models that do not contain any Turing complete reals. begingroup Would it be the same process as proving that a sequence is Cauchy, just instead of lt epsilon, I find that it is gt epsilon Im not familiar with showing that sequences arent Cauchy as Ive almost exclusively dealt with showing that sequences are Cauchy. |x_p-x_q|<2^_0$ and to constructive systems with various forms of compactness. If fx ngis a Cauchy sequence in E, show that ff(x n. This contradicts the de nition of uniform continuity for ' 3. Consider the sequences x n 1 n y n 1 2n: Then jx n y nj 12n<1n:On the other hand, jg(x n) g(y n)j 1 y2 n 1 x2 n 3n2 >3 if n>1.The greatest variety seems to be in Dedekind-style approaches: in addition to "the" Dedekind real numbers which satisfy locatedness (for all $qn. Solution: The function is not uniformly continuous. A double sequence x ( x jk ) is said to be strongly almost convergent to a number L if. The idea of strong almost convergence for single sequences is due to Maddox 64, and for double sequences, to Baarir 12. Obviously, every weakly p-convergent sequence is weakly p-Cauchy, and the weakly -Cauchy sequences are precisely the weakly Cauchy sequences. As in case of single sequences 63, Theorem 2.5 gives the equivalence of these two definitions. In constructive mathematics there are many possible inequivalent definitions of real numbers. is weakly p-Cauchy if for each pair of strictly increasing sequences (kn)n and (jn)n of positive integers, the sequence (xkn xjn)n is weakly p-summable in X.
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