integer $N_n$, which depends only on $N$ and $n$ such that $\lambda(\alpha) \leq N_n$ for all $\alpha \in H^n(K,
\mu_\ell^{\otimes n})$.

Since $N$ is $(2, \ell)$-uniform bound for $K$, by ( \[lemma:uniform\_bound\_d\_d+1\]), $N$ is an $(n, \ell)$-uniform bound for $K$ for all $n \geq 2$. Let $\alpha \in H^n(K, \mu_\ell^{\otimes
   n})$. Then, by (\[theorem:krashen1\]), $\lambda(\alpha)$ is bounded in terms of eind$(\alpha)$, $N$ and $n$. Since eind$(\alpha) \leq N$, $\lambda(\alpha)$ is bounded in terms of $N$ and $n$.

\[cor:uniformbound\_uinvariant\] Let $K$ be a field of characteristic not equal to 2. Suppose that there exists an integer $M$ such that for all finite extensions $L$ of $K$, $H^M(K,
  \mu_2)  = 0$ and $N$ is a $(2,2)$-uniform bound for $K$. Then there exists $N'$ which depends only on $N$ and $M$ such that for any finite extension $L$ of $K$, $u(L) \leq N'$.

Since the conditions on $K$ are also satisfied by any finite extensions of $K$, it is enough to find an $N'$ which depends only on $N$ and $M$ such that $u(K) \leq N'$.

Since $N$ is a $(2,2)$-uniform bound for $K$, by (\[cor:symbol\_length\]) there exist integers $N_n$ for $1 \leq n
  < M$, which depends only on $N$ and $n$ 