How to prove that this inequality holds

Question

Let $A$ be a unital Banach algebra.

I wanted to prove the following inequality but didn't manage:

$$ \begin{align} \left | \|a\| - \inf_{d \in A: \|d\| = 1}\|bd\| \right | \le \inf_{\|d\|=1} \left | \|a\| - \|bd\|\right | \end{align}$$

Here $a \in A$. Now I am starting to doubt that it holds but I could also not find a counterexample. Could somebody either help me prove it or provide a counterexample? Thanks.

Answer 1

For every $d\in A$ such that $\|d\|=1$ one has $$\zeta(a)\leq\|bd\|+\|a-b\|.$$

So $$\zeta(a)\leq\zeta(b)+\|a-b\|.$$

Using a symmetric reasoning we get $$\zeta(b)\leq\zeta(a)+\|a-b\|,$$

and the inequality is proved.