We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$, where $\mathcal{D}$ is a distribution over $\{1, 2, \ldots, U\} \subset \mathbb{N}$ with cumulative distribution function $F_\mathcal{D}$.

We define the subcollection that includes only the values in $\boldsymbol{S}$ that are above $Q(p)$, where $Q$ is the quantile function. That is:

$$ \boldsymbol{S}_{\geq p} \overset{\small \text{def}}{=} \left\{X : X \in \boldsymbol{S} \text{ and } p\leq F_{\mathcal{D}}(X)\right\} $$

(in words: $X \in \boldsymbol{S}_{\geq p}$ if and only if it $p$ proportion of the population are smaller or equal to it)

(below we mark $\pmb{\sum}\boldsymbol{C}$ as the sum of all elements in collection $\boldsymbol{C}$)

Is the following true?

$$ \forall n,\; \mathbb{E}\left[\frac{\pmb{\sum}\boldsymbol{S}_{\geq p}}{\pmb{\sum}\boldsymbol{S}}\right] \le \frac{\mathbb{E}\pmb{\sum}\boldsymbol{S}_{\geq p}}{\mathbb{E}\pmb{\sum}\boldsymbol{S}} $$

**note** this is an extracted step of a different question expectation of upper quantile proportion.
This question is simpler and more focused. (+ if the statement above is true the previous question is also solved)