- #1

#### playboy

## Homework Statement

Let [itex]a_{0} = a >1 [/itex] and let [itex]a_{n+1} = a^{a_n}[/itex].

Show that {[itex] a_{n} [/itex]} comverges for [itex] a < e^{e^-1} = 1.4446678 [/itex]

## Homework Equations

This is a Theorm I learned in Real Analysis and hope to apply it to this problem:

Theorm: If a sequence is montonically increasing and bounded, then it is convergent

## The Attempt at a Solution

{[itex] a_{n} [/itex]} = {[itex]a, a^a, a^{a^a}, ...[/itex]}

Clearly, {[itex] a_{n} [/itex]} is monotonically increasing is is bounded below by a.

How do I show that it is bounded above?

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