I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process"


In one of the exercises, exercise 8.9 in page 133, it is asked to prove continuity on $D_{M}$ for the function defined for paths $\alpha \in D_{M}$ as:

$G(\alpha )= \sup_{t \geq 0} e^{-t} d_{M} (\alpha(t), \alpha(t-) ) $

I already have a couple of weeks thinking about it, and even though there is a hint for this exercise I have been unable to solve this. Any recommendation ? My silly approach goes as follows: Take $\alpha \in D_{M}$ and a sequence $\alpha_{n} \in D_{M}$ such that $s (\alpha_{n}, \alpha ) \rightarrow 0$. Then Assuming that the path $\alpha$ is not continuos: \begin{align} & \quad | \sup_{t \geq 0} e^{-t} d_{M} (\alpha(t), \alpha(t-) ) - \sup_{t \geq 0} e^{-t} d_{M} (\alpha_{n}(t), \alpha_{n}(t-) ) | \\ &= |e^{-T}d_{M} (\alpha(T), \alpha(T-) ) - e^{-T_{2}} d_{M} (\alpha_{n}(T_{2}), \alpha_{n}(T_{2}-) ) | \\ &\leq |e^{-T}d_{M} (\alpha(T), \alpha(T-) ) - e^{-T} d_{M} (\alpha_{n}(T), \alpha_{n}(T-) ) | \\ &= e^{-T} |d_{M} (\alpha(T), \alpha(T-) ) - d_{M} (\alpha_{n}(T), \alpha_{n}(T-) ) | \end{align}

I know its lame but I really do not have idea how to proceed. Any suggestions would be great. Thanks in advance


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