I was reading "The Theory of Matrices" by Felix Gantmacher and found that he uses the concepts of continuity and limits with matrices which was something new for me. Specifically he used the following argument to prove the Sylvester's Determinant Identity: $A$ is an $n$ by $n$ square matrix.
The questions:
1- What does $\epsilon E$ represent? What are $\epsilon^j + \dots$? And how can I choose a sequence $\epsilon_m \rightarrow 0 $?
2- May you please provide me with some resources where I can learn using those concepts (continuity, limits, and sequences) with matrices? (online lectures are preferred)
1-I am going to assume that $E$ represents the identity matrix. Then $A_\epsilon = A+\epsilon E$ is simply adds $\epsilon$ to the diagonal entries of $A$. The minor of the first $j$ rows and columns of $A_\epsilon$ will be a polynomial in $\epsilon$ of degree $j$ whose leading coefficient is $1$ and whose remaining coefficients are entries in $A$. (In particular, this minor is the product of $j$ diagonal entries $a_{ii}+\epsilon$ plus some terms of lower degree in $\epsilon$.) This polynomial has finitely many positive roots, $\epsilon^{(j)}_1, \dots, \epsilon^{(j)}_{k_j}$. The collection of all these positive roots $\epsilon^{(1)}_1, \dots, \epsilon^{(1)}_{k_1}, \dots, \epsilon^{(p)}_1, \dots, \epsilon^{(p)}_{k_p}$ for $j=1,\dots,p$ is still finite.
Thus we can make a sequence of positive $\epsilon_m \rightarrow 0$ where each $\epsilon_m$ is smaller than every one of these positive roots $\epsilon^{(j)}_{i}$. This is a standard real analysis observation, but to be concrete, since the above collection is finite, you can choose some positive integer $N$ so that $1/N$ is smaller than all of them and set $\epsilon_m = \frac{1}{N+m}.$
2-Any resource on basic real analysis covering continuity and sequences will serve you. Understanding Analysis by Stephen Abbott is very friendly and readable, but there are countless others. There's nothing really special about the use of matrices here.