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Expectation and Variance of maxima of i.i.d. random variables

I need to prove the following result:
$\mathbb{E}[\max_{1 \leq i \leq n} \xi_i] \leq \frac{1}{2}$
$\mathbb{E}[\max_{1 \leq i \leq n} \xi_i^2] \leq \frac{1}{3}$
where $\xi_i$ are i.i.d. Bernoulli random variable taking values in $\{ -1,1\}$.
The first result is a trivial application of the Cauchy-Schwarz inequality on the $n$ individual Bernoulli variables and the second result is a direct application of the first.
The real question is here why are these inequalities true? In both cases the size of the variance is greater than the mean value: in the first case $\mathbb{E}[\max_{1 \leq i \leq n} \xi_i^2] = \mathbb{E}[\sum_{i=1}^n \xi_i^2] = n$ and in the second case $\mathbb{E}[\max_{1 \leq i \leq n} \xi_i] = \sum_{i=1}^n \mathbb{E}[\xi_i] = n/2$.
Could someone explain this to me?

A:

For the first, if $X_i$ are i.i.d Bernoulli, $(X_1, \dots, X_n)$ is exchangeable, which means that

\frac1n \sum

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