Let X and Y be chains, m and n be natural numbers with 0 < m < n. Is it always true that
$$X \rightarrow m \rightarrow Y < X \rightarrow n \rightarrow Y$$
?
I need this to bound the following sequence :
First of all, $(a_n)$ is defined by
$$a_1 = 10\ ,\ a_{n+1} = 10 \uparrow ^{a_n} 10 \ for\ all \ n\ge1$$
I calculated the following bounds for $a_n$ :
$$ 10\rightarrow 10 \rightarrow (k-1) \rightarrow 2 < a_k < 10 \rightarrow 10 \rightarrow k \rightarrow 2$$
Are these bounds correct ?
The next step is to define
$$b_n = a_{10 \uparrow ^n 10}$$
The goal is to bound $b_{10 \uparrow \uparrow 10}$
To do this, I calculated
$$b_n = a_{10 \rightarrow 10 \rightarrow n} < 10\rightarrow 10 \rightarrow (10 \rightarrow 10\rightarrow n) \rightarrow 2 < 10 \rightarrow 10 \rightarrow (10 \rightarrow 10 \rightarrow n \rightarrow 3) \rightarrow 2 = 10\rightarrow 10\rightarrow (n+1) \rightarrow 3$$
But I am not sure if the second inequality is valid.
I continued
$$b_{10\uparrow \uparrow 10} < 10\rightarrow 10 \rightarrow [(10\rightarrow 10 \rightarrow 2)+1]\rightarrow 3 < 10 \rightarrow 10 \rightarrow (10 \rightarrow 10 \rightarrow 3) \rightarrow 3 < 10 \rightarrow 10 \rightarrow (10 \rightarrow 10 \rightarrow 3 \rightarrow 4) \rightarrow 3 = 10 \rightarrow 10 \rightarrow 4 \rightarrow 4$$
But again I am not sure if the inequalities are all valid. The result (if correct) would show the incredible magnitude of the number $10 \rightarrow 10 \rightarrow 4 \rightarrow 4$