In a proof, it is assumed that for $x=(\alpha, 0, ..., 0)'$ with $\alpha \in R_+$ that the norm $||{x}||\leq\alpha$
The norm is not further specified.
However, I can define a norm which is , for example $|| \cdot||=2*|| \cdot||_2$, where the above does not hold.
Is my reasoning correct? And if so, is there an assumption I can make (without specifying the norm) such that $||{x}||\leq\alpha$ holds?
Yes, you are right.
Note that if assume that $\|\cdot\|=\|\cdot\|_p$ for some $p\in[1,+\infty]$, then that inequality holds.