Bound on Norm of Power of Element in Normed Algebra

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.

Let $x \in A$ and $n \in \N$.

Then:


 * $\norm {x^n} \le \norm x^n$

Proof
The proof proceeds by induction.

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:


 * $\norm {x^n} \le \norm x^n$

Basis for the Induction
We have:


 * $\norm {x^1} = \norm x = \norm x^1$

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:


 * $\norm {x^k} \le \norm x^k$

from which it is to be shown that:


 * $\norm {x^{k + 1} } \le \norm x^{k + 1}$

Induction Step
This is the induction step.

We have: