  For a symmetric convex body $K\subset\mathbb{R}^n$, the Dvoretzky dimension
$k(K)$ is the largest dimension for which a random central section of $K$ is
almost spherical. A Dvoretzky-type theorem proved by V.~D.~Milman in 1971
provides a lower bound for $k(K)$ in terms of the average $M(K)$ and the
maximum $b(K)$ of the norm generated by $K$ over the Euclidean unit sphere.
Later, V.~D.~Milman and G. Schechtman obtained a matching upper bound for
$k(K)$ in the case when $\frac{M(K)}{b(K)}>c(\frac{\log(n)}{n})^{\frac{1}{2}}$.
  In this paper, we will give an elementary proof of the upper bound in
Milman-Schechtman theorem which does not require any restriction on $M(K)$ and
$b(K)$.
