Every index in a representation of $n/2^n$ exceeds $n$

Write $\operatorname{Rep}$ for a representation $\frac{n}{2^n}=\sum_k \frac{a_k}{2^{a_k}}$ by pairwise distinct positive integers $a_k$ (Erdős Problem 261). Theorem. If $t\ge2$ and $\frac{n}{2^n}=\sum_{k=1}^t\frac{a_k}{2^{a_k}}$ with positive $a_k$, then $a_k>n$ for every $k$.

This is the first normalisation step for Problem 261: since $x\mapsto x/2^x$ is nonincreasing on the positive integers, a single index $a\le n$ already contributes at least $n/2^n$, leaving no room for the other (positive) terms.

Formalization Note. Injectivity of $a$ is assumed (to match Erdos261Prop) but not used.