Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\chi_n$ be the characteristic polyomial of $A_n$ and $\chi=a\prod(X-\lambda_i)$ be that of $A$, then $\chi_n\mathop{\to}_{n\rightarrow\infty}\chi$.

If a sequence of polynomials converges to another, then by continuity the roots of the polynomials in the sequence must converge to the roots of the limit polynomial (with the same multiplicity). Hence the eigenvalues of the sequence of matrices indeed convergence to the eigenvalues of $A$.

Likewise, since any submatrix of $A_k$ converges to the corresponding submatrix of $A$, that is sufficient to say that $\delta_s(n)\to\delta_s$.

The answer of @Hippalectryon is morally correct. However, there is this problem of approaching a polynomial with multiple roots, somehow delicate. $\tiny{\text{Can be proved with the argument principle from complex analysis.}}$ We'll work directly with the eigenvalues, bypassing the characteristic polynomial.

Use this standard observation: if two symmetric matrices are ordered $B\prec C$ then their vectors of eigenvalues ( ordered increasingly) also satisfy $\lambda(B) \prec \lambda(C)$ ( that is $\lambda_i(B) \le \lambda_i(C)$ for all $i$).

Now, if if $A_n \to A$ then for any $\epsilon >0$ we have $A- \epsilon I \prec A_n \prec A + \epsilon I$ for $n \ge n_{\epsilon}$, and so $$\lambda_i(A) - \epsilon \le \lambda_i(A_n) \le \lambda_i(A) + \epsilon$$

$\tiny{\text{(the question for the max for submatrices is simple: $\lim$ and $\max$ commute ).}}$

$\bf{Added:}$ I am giving this " continuity of the roots" question more thought. Of course, @Hippalectryon is right. But how do we truly convince ourselves that the roots of $P_n$ approach the roots of $P$? First, a hands-on approach: The roots of $P_n$ will all lie in a bounded region since the coefficients of $P_n$ are all bounded. If the roots of $P_n$ did not approach those of $P$ then we would find, by compactness, a subsequence whose roots approach some other $n$-uple. But that would mean in the limit that $P$ would decompose using that $n$-uple, that is, it would have two distinct decompositions, contradiction. The high-brow explanation is that the map roots $\mapsto$ polynomials from $\mathbb{C}^n$ to $\mathbb{C}^n$ is continuous and proper and induces a bijective map $\mathbb{C}^n/S_n \to\mathbb{C}^n$, which is bijective, continuous and closed, hence a homeomorphism. $\tiny{\text{(surjectivity is equivalent to the fundamental theorem of algebra)}}$