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'To obtain finite results for an amplitude involving divergent diagrams, we have so far used the following procedure: Compute the diagrams using a regulator to obtain an expression that depends on the bare mass (m0), the bare coupling constant (e0) and some UV cutoff ([itex]\Lambda[/itex]). Then compute the physical mass (m) and the physical coupling constant (e), to whatever order is consisten with the rest of the calculation; these quantities will also depend on m0, e0 and [itex]\Lambda[/itex]. To calculate an S-matrix element one must also compute the field-strength renormalization Z (according to the LSZ formula). Combining all of these expression, eliminate m0 and e0 in favour of m and e; this step is the 'renormalization'. The resulting expression for the amplitude should be finite in the limit [itex] \Lambda \to \infty[/itex].'

Where does the author actually use this procedure to compute an S-matrix element?

Have I understood it correctly if the point is that when one inserts the relations m(m0), e(e0) and Z into the LSZ formula the divergences in the respective relations cancel each other out to a given order?