User blog:Simply Beautiful Art/Lower bound on Down-arrow notation

Down-arrows are defined as

$$ a\downarrow^{1}b=a^b $$

$$ a\downarrow^{n}1=a $$

$$ a\downarrow^{n}b=(a\downarrow^{n}(b-1))\downarrow^{n-1}a $$

Compared to the up-arrow notation defined by

$$ a\uparrow^{1}b=a^b $$

$$ a\uparrow^{n}1=a $$

$$ a\uparrow^{n}b=a\uparrow^{n-1}(a\uparrow^{n}(b-1)) $$

On the down-arrow page (http://googology.wikia.com/wiki/Down-arrow_notation), a few lower bounds are given. Let's extend them to the more general lower bounds, given large enough b:

$$ a\downarrow^{2n}b > a\uparrow^{n}(a(b-1)) $$

$$ a\downarrow^{2n-1}b > a\uparrow^{n}b $$

which is a fairly straightforward induction problem.