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 $$A$$.

The $$1$$-norm of a matrix is usually the max column sum (that is, the max column $$\ell_1$$-norm). Are you sure that your book defines it as you said?
If it is the max column sum, then the norm will be $$|\lambda_j|$$ plus off-diagonal elements. The off-diagonal elements are bounded by $$\epsilon,$$ and the spectral radius will bounded any $$|\lambda_j|$$, by definition.