So this is my attempt at a proof, and there are a bunch of inequalities, which always confuse me

, so let me know if there are any mistakes.

Let

as usual. I will use

k (instead of

n) to avoid confusion with

.

Lemma 1 (Knoebel).

iff

for all

.

Lemma 2.

for all real

.

Proof.

The base-

tetrational function is continuous and monotonic?

Lemma 3.

for all positive real

and real

.

Proof.

The function

for all

, thus

. If

, then

, so

.

Lemma 4 (lower bound). For all integer

,

.

Proof.

Since

, then obviously

. Together with lemma (2), this implies that

. Substituting

(hypothesis), this can be written as

which implies

by lemma (1).

Lemma 5 (decreasing). For all integer

and

,

.

Proof.

We have

for all integer

by definition. It follows that

by lemma (3). Thus

which implies

by lemma (1).

Theorem.

Proof.

for all

by lemma (4) and lemma (5).

... In the limit, the squeeze theorem and completeness should guarantee that the limit exists and converges to

.

Is this right?

Andrew Robbins