I am confused on the Lemma 5.7 (Lipschitz maximal inequality) here. Let me first restate the definition and the lemma:

Def $\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that $$|X_t-X_s|\leq Cd(t,s),\text{ for all }t,s\in T.$$

Lemma Suppose $\{X_t\}_{t\in T}$ is a Lipschitz process and is $\sigma^2$-subgaussian for every $T$. Then $$E[\sup_{t\in T}X_t]\leq \inf_{\epsilon>0}\{\epsilon E[C]+\sqrt{2\sigma^2\log N(T,d,\epsilon)}$$

To prove the lemma, we split $X_t$ into two parts, i.e. $$\sup_{t\in T}X_t\leq \sup_{t\in T}\{X_t-X_{\pi(t)}\}+\sup_{t\in T}X_{\pi(t)}$$ then take expectation we have $$E[\sup_{t\in T}X_t]\leq \epsilon E[C]+\sqrt{2\sigma^2\log|N|}$$

I am a bit confused on the applicability of this lemma on random process that its $E[\sup X_t]$ is negative.

In the proof, we use the Lipschitz bound to control $\sup_{t\in T}\{X_t-X_{\pi(t)}\}\leq Cd(t,\pi(t))$. Clearly, this bound $Cd(t,\pi(t))$ is non-negative, thus the the resulting bound $ \inf_{\epsilon>0}\{\epsilon E[C]+\sqrt{2\sigma^2\log N(T,d,\epsilon)}$ in the lemma is also non-negative.

But, what if the expectation of sup of random process is negative? Then the bound given by the lemma is very loose.

For example, I guess this process $\sum_{i=1}^n(-100+w_i)sin(\theta_i)$ where $w_i$ is iid standard gaussian would have negative expectation of supreme.

Am I misunderstood something on this lemma?