The following problems can be found in chapter 8 of the PDE book by Evans. Here $U$ is an open, bounded set in $\mathbb{R}^n$ with smooth boundary. After problem 4, I was asked to solve problem 5. To take the first step, one can consider $x_0$ to be a regular value of $\mathbf{u}$. And, by employing the inverse function theorem, one can show that $$\deg(\mathbf{u},x_0)=\sum_{\mathbf{u}^{-1}(\{x_0\})}\mathrm{sgn}(\det D\mathbf{u}(x)),$$ upon using integration by substitution as below. Note that this is a finite sum. Next we have to consider the general case in which $x_0$ may not be regular. It's desirable to use Sard's theorem, from which we conclude that regular values constitute a dense set. Thus, we are able to find a sequence of regular values that converges to $x_0$. But what's next? Is there any theorem to guarantee that the sequential limit exists? I need someone to help me out. Please. Thank you very much.
The standard approach here is to invoke Sard's theorem. First do the proof assuming that the Jacobian does not vanish at any point $y\in U$ with $\mathbf{u}(y) = x_0$. The set of $x_0$ for which this holds is called the set of regular values of $\mathbf{u}$, and Sard's theorem says that the regular values of $\mathbf{u}$ are dense in the image of $\mathbf{u}$. So once you have the result for regular values, you can recover the result for non-regular values by a limiting argument, as the limit of an integer sequence is an integer.