Recursive set for a string

Question

Given the definition:

Let Σ be the alphabet defined as follows:  Σ = {e, l, v} 
We now define the set of strings P according to the following: 
BASIS STEP: If x ∈ Σ then x ∈ P 
RECURSIVE STEP: If w ∈ P and x ∈ Σ, then xwx ∈ P

How to show that the string level is in P.

I do understand what is being asked and that I must use the recursive step to produce the target string. But I just don't know exactly how to go about it in a way that is efficient and accurate. Any help with an explanation is greatly appreciated.

Answer 1

How about

$$ v \in \Sigma \Longrightarrow v \in P $$ and $$v \in P \land e \in \Sigma \Longrightarrow eve \in P $$ so $$ eve \in P \land l \in \Sigma \Longrightarrow \boxed{level \in P}$$