EDB — 02F

[02F]Prerequisites:[1Z7]↺↻.Let \(a_ n,b_ n\) be real sequences (which can have variable signs, take value zero, and are not necessarily infinitesimal); let \(X=ℝ^ℕ\) the space of all sequences.

Recall that the notation \(a_ n=O(b_ n)\) means:

Show these results:

• for \(a,b∈ X , a=(a_ n)_ n,b=(b_ n)_ n\) consider the relation

  \[ aRb \iff a_ n=O(b_ n) \]

  prove that \(R\) is a preorder;

• define \(x ≍ y\iff (xRy ∧ yRx)\) then \(≍\) is an equivalence relation, \(R\) is invariant for \(≍\), and the projection \(⪯\) is an order relation on \(X/≍\) (hint: use the Prop. [1Z7]↺↻).

• Define (as usually done)

  \[ \hat a≺ \hat b \iff (\hat a⪯ \hat b ∧ \hat a≠ \hat b) \]

  for \(\hat a,\hat b∈ X/≍\), \((a_ n)_ n\in \hat a,(b_ n)_ n\in \hat b\) representatives; assuming \(b_ n≠ 0\) (eventually in \(n\)), prove that

  \[ \hat a≺ \hat b \iff 0=\liminf _ n \frac{a_ n}{b_ n}≤ \limsup _ n\frac{a_ n}{b_ n} {\lt}∞\quad . \]

The above discussion is related to Definition 3.2.3 (and following) in [ 3 ] .