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ar.grig
ar.grig
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We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\alpha<\beta: \lim_{n\to\infty}\frac{y_{\beta n}}{y_{\alpha n}}=0$. Can we prove that for any infinitesimal sequence $y$ there exists $y_\alpha$ such that $\lim_{n\to\infty}\frac{y_{\alpha n}}{y_{n}}=0$?

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences of rational numbers with the property: $\forall\alpha<\beta: \lim_{n\to\infty}\frac{y_{\beta n}}{y_{\alpha n}}=0$. Can we prove that for any infinitesimal sequence $y$ there exists $y_\alpha$ such that $\lim_{n\to\infty}\frac{y_{\alpha n}}{y_{n}}=0$?

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\alpha<\beta: \lim_{n\to\infty}\frac{y_{\beta n}}{y_{\alpha n}}=0$. Can we prove that for any infinitesimal sequence $y$ there exists $y_\alpha$ such that $\lim_{n\to\infty}\frac{y_{\alpha n}}{y_{n}}=0$?

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ar.grig
ar.grig
  • 1.1k
  • 5
  • 10

Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences of rational numbers with the property: $\forall\alpha<\beta: \lim_{n\to\infty}\frac{y_{\beta n}}{y_{\alpha n}}=0$. Can we prove that for any infinitesimal sequence $y$ there exists $y_\alpha$ such that $\lim_{n\to\infty}\frac{y_{\alpha n}}{y_{n}}=0$?