# Criterion for the Existence of a Bounded Left Approximate Identity

I am reading through the text "Complete Normed Algebras" by F.F. Bonsall and John Duncan and I'm having some troubles understanding proposition 2 in chapter 1, section 11 (page 58) regarding the existence of a bounded left approximate identity for a normed algebra.

The definition I am using for bounded left approximate identity is:

Definition: Let $A$ be a normed algebra. A bounded left approximate identity is a net $\{ e(\lambda) \}_{\lambda \in \Lambda}$ such that $e(\lambda)x \to x$ for every $x \in A$ and there exists a $K > 0$ such that $\| e(\lambda) \| \leq K$ for all $\lambda \in \Lambda$.**

Proposition 2 on page 58 states:

Proposition 2: Let $A$ contain a bounded set $U$ such that, given $x \in A$, $\epsilon > 0$ there exists $u \in U$ with $\| x - ux \| < \epsilon$. Then $A$ has a bounded left approximate identity.

For $\epsilon > 0$, it is shown in the proof that for any finite subset $F$ of $A$ there exists a $w \in W = \{ v \circ u : v, u \in U \}$ such that:

$$\| x - wx \| < \epsilon$$

Where $v \circ u := v + u - vu$ is the quasi-product of $v$ and $u$. The authors prove this by induction (which is fairly straightforward to follow).

The authors then state that "the result is then clear by consideration of the product directed set $\mathbb{N} \times F(A)$ where $F(A)$ denotes the directed set of all finite subsets of $A$."

I am not sure what this means, perhaps because I am not that familiar with nets. Explicitly, what net becomes the bounded left approximate identity for $X$? Any help would be greatly appreciated!