Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that


$$|f|\_{var}= \sum\_{n=1}^{\infty} var\_n f < \infty,$$


where $var\_n f = sup\{|f(x)-f(y)|: x,y \in X , x\_k = y\_k, \forall k = 1,\dots, n\}$.
Also, suppose that 
$\sum\_{i =1}^\infty \exp(\sup\_{x \in [i]}f(x)) < \infty$, where $[i] = \{x \in X: x\_1 = i\}$. We want to conclude that for $t>1$ exists a constant for $C\_t$ such that


$$\sup\_{i \in I} \sup\_{x \in [i]}|f(x)| \exp( (t-1) sup\_{x \in [i]}f(x)) < C\_t.$$


To be more precise, I am trying to understand the proof of the Lemma 3.1 of the I. Morrison's following paper: "[Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type](http://personal.maths.surrey.ac.uk/st/I.Morris/entropy3.pdf)". But I think that understand this less general case, I understand the paper by myself. Thank you very much.



For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) sup_{x \in [i]}f(x) )$ is bounded
I have just found out that is very simple.


We have that $\sup f - \inf f \le \text{var}\_1f$ and also that $\sup\_{x \in [i]}f(x) \to - \infty $ as $ i \to \infty$.


Then, for $i$ sufficiently large
\begin{align}
\sup\_{x \in [i]}|f(x)| \exp( (t-1) sup\_{x \in [i]}f(x))&
= \sup\_{x \in [i]}(-f(x)) \exp( (t-1) sup\_{x \in [i]}f(x))\\
&\le \left(\text{var}\_1 f+ \inf\_{x \in [i]}(-f(x)) \right) \exp( (t-1) sup\_{x \in [i]}f(x))\\
&= \left(\text{var}\_1 f + \sup\_{x \in [i]}f(x) \right) \exp( (t-1) sup\_{x \in [i]}f(x)).
\end{align}


Hence, taking $i \to \infty$, the limit vanishes and the boundedness is guaranteed. 


