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

https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf

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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