Question on the definition of a norm of a linear transformation

Question

I am working through chapter 5 of Rudin's Real and Complex Analysis, and there is something about the norm of a linear transformation I don't understand. Rudin defines the norm of a linear transformation $\Lambda$ from a normed linear space X to a normed linear space Y as follows: $$\lvert\lvert\Lambda\rvert\rvert = \sup\{\lvert\lvert\Lambda x\rvert\rvert:x\,\epsilon\, X,\Vert x\Vert \le 1\}$$ Rudin then points out that since $\Lambda$ is linear, we have $$ (1)\,\,\, \Vert \Lambda(\alpha x)\Vert = \Vert\alpha\Lambda x\Vert=\vert\alpha\vert\Vert\Lambda x\Vert$$ So far so good. Everything makes sense up to this point. But then Rudin states that (1) implies that we can take the supremum over all $x\,\epsilon\,X $ such that $\Vert x \Vert=1$ rather than over all $x\,\epsilon\,X $ such that $\Vert x \Vert\le1$. It is this part that I don't understand. Can someone explain how this conclusion is reached from (1)?