  Let $A = K[X_1,...,X_n]$ and let $I$ be a graded ideal in $A$. We show that
the upper bound of Multiplicity conjecture of Herzog, Huneke and Srinivasan
holds asymptotically (i.e., for $I^k$ and all $k \gg 0$) if $I$ belongs to any
of the following large classes of ideals: \begin{enumerate}[\rm (1)]
  \item radical ideals. \item monomial ideals with generators in different
degrees. \item zero-dimensional ideals with generators in different degrees.
\end{enumerate} Surprisingly, our proof uses local techniques like analyticity,
reductions, equimultiplicity and local results like Rees's theorem on
multiplicities.
