04/12/2009, 08:58 AM

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

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