Resolvent Mapping is Continuous/Banach Algebra/Lemma

Lemma
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.

Let $x \in A$. Define $S : \C \to A$ by:
 * $\map S \lambda = \lambda {\mathbf 1}_A - x$

Then $S$ is continuous.

Proof
We have, for $\lambda, \mu \in \C$:

Let $\epsilon > 0$ and $\lambda, \mu \in \C$ be such that:
 * $\cmod {\lambda - \mu} < \epsilon$

Then, we have:
 * $\norm {\map S \lambda - \map S \mu} < \epsilon$

Hence $S$ is continuous.