On page 1290 of Crandall 1978, there is a line saying
From Corollary (7.1) and Lemma (7.4) we have $$m(2^{A_k}-3^k)<\sum_{j=0}^{k-1}(3+\frac{1}{m})^{k-1}$$
But looking at Corollary (7.1) we have $$m(2^{A_k}-3^k)=\sum_{j=0}^{k-1}2^{A_j}3^{k-1-j}$$
and looking at Lemma (7.4) we have $$2^{A_j}\leq(3+\frac{1}{m})^{j}$$
So for me (7.1) and (7.4) would lead to $$m(2^{A_k}-3^k)<\sum_{j=0}^{k-1}(3+\frac{1}{m})^{j}3^{k-1-j}$$
but RHS is $$3^{k-1}\sum_{j=0}^{k-1}(1+\frac{1}{3m})^{j}=3^{k-1}\frac{(1+\frac{1}{3m})^{k}-1}{(1+\frac{1}{3m})-1}=m((3+\frac{1}{m})^{k}-3^k)$$
So it leads to $$m(2^{A_k}-3^k)<m((3+\frac{1}{m})^{k}-3^k)$$ $$2^{A_k}<(3+\frac{1}{m})^{k}$$ or (7.4) again
Where does the first line came from? Am I not looking at the right place in those Lemma?