Let $X_1(t),...,X_N(t)$ be $N$ independent, mean zero random processes indexed by points $t \in T$. Let $\epsilon_1,...,\epsilon_N$ be independent symmetric Bernoulli random variables. Prove that
$$ \frac{1}{2} \mathbb{E} \sup_{t \in T} \sum_{i=1}^N \epsilon_i X_i(t) \leq \mathbb{E} \sup_{t \in T} \sum_{i=1}^N X_i(t) \leq 2 \mathbb{E} \sup_{t \in T} \sum_{i=1}^N \epsilon_i X_i(t) $$
This is the problem from the high dimensional probability 7.1.9(Vershynin)
I'm trying to solve this problem and already show the upper bound.
To show the lower bound,
\begin{align*} \mathbb{E} \sup_{t \in T} \sum_{i=1}^N \epsilon_i X_i(t) &= \mathbb{E} \sup_{t \in T} \sum_{i=1}^N \epsilon_i [X_i(t) + \mathbb{E} -X_i'(t))] \\ &\leq \mathbb{E} \sup_{t \in T} \sum_{i=1}^N \epsilon_i [X_i(t) - X_i'(t)] \\ &= \mathbb{E} \sup_{t \in T} \sum_{i=1}^N [ X_i(t)- X_i'(t)] \end{align*} To end the proof of the lower bound, I think I need to show that \begin{align} \mathbb{E} \sup_{t \in T} \sum_{i=1}^N [ X_i(t)- X_i'(t)] \leq \mathbb{E} \sup_{t \in T} \sum_{i=1}^N X_i(t)+ \mathbb{E} \sup_{t \in T} \sum_{i=1}^N X_i'(t). \;\;\;\;\;\;(1) \end{align}
$X_i'(t)$ is the independent copy of the random process $X_i(t)$
Does (1) inequality hold?