Spectral radius is the greatest lower bound for some matrix norm

I’m studying matrix analysis with Horn and Johnson’s book.

I have something trouble while reading the book.

There is lemma 5.6.10 lemma and the following is the proof of that Proof of lemma.

I have trouble in two lines below from the matrix such that 1-norm of (D_t \triangle D_t^{-1}) is less and equal to (\rho(A)+\epsilon).

1-norm is defined as the sum of all element in the matrix.

I understood that off-diagonal elements can be bounded by epsilon for large t. However, I cannot understand how does the sum of absolute values of eigenvalues will be bounded by spectral radius of A.