In the book Principles of mathematical analysis by Walter Rudin,He writes:

"For $ A\in L(\Bbb R^n,\Bbb R^m)$, define the norm $||A||$ of $A$ to be sup of all numbers $|Ax|$, where $x$ ranges over all vectors in $\Bbb R^n$ with $|x|\le1$.
Observe that the inequality
$$ |Ax|\le||A|||x| $$
holds for all $ x\in \Bbb R^n$. Also, if $\lambda$ is such that $|Ax|\le \lambda|x| $ for all $ x\in \Bbb R^n $, then $||A|| \le \lambda $."

Here I am not able to understand the last line. where he says $||A|| \le \lambda$.

Since $ |Ax|\le||A|||x| $ and $|Ax|\le \lambda|x|$ both hold for all $ x\in \Bbb R^n $, then why necessarily $||A|| \le \lambda$ ?