Before we proceed any further, let us explore how we can get rid of ε transitions. Let consider an ε transition as follows: T(u,ε) = w. This means once the machine is at state u, it automatically transitions to w.
We can modify the machine M = (S,Σ,T,s,A) as follows. First, we remove state w from S, then we change all T(x,σ) = w to become T(x,σ) = u for all x ∈ S and σ ∈ Σ. We repeat this process until there is no ε transitions remaining.