Implications

We have the following indirect implication of form equivalence classes:

74 \(\Rightarrow\) 74
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
74 \(\Rightarrow\) 74 The Axiom of Choice, Jech, 1973b, page 21

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
74:

For every \(A\subseteq\Bbb R\) the following are equivalent:

  1. \(A\) is closed and bounded.
  2. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A.
Jech [1973b], p 21.

74:

For every \(A\subseteq\Bbb R\) the following are equivalent:

  1. \(A\) is closed and bounded.
  2. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A.
Jech [1973b], p 21.

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