Let $A$ and $B$ be Banach Algebras and $T: A \rightarrow B$ be a homomorphism.
Using the inclusion $T(A^{-1}) \subseteq B^{-1}$, prove $\sigma(T(a), B) \subseteq \sigma(a,A)$ for all $a \in A$.
I think I understand why this is true and this was my idea for a proof but it feels kind of wrong, especially using cardinality:
For this proof let $|X|$ denote the size of the set $X$.
$T(A)^{-1} \subseteq B^{-1}$
therefore $|A^{-1}| \leq |B^{-1}|.$
therefore $|\sigma(a,A)| \geq |\sigma(T(a))|$
and additionally, $T(A^{-1}) \subseteq B^{-1} \Rightarrow \sigma(a,A) \cap \sigma(T(a),B) \subseteq \sigma(a,A)$
and thus $\sigma(T(a), B) \subseteq \sigma(a,A)$ for all $a \in A$