defining Mixed integer linear inequalities for a set of variables

The problem is described as follows: considering $$n$$ variables which are continuous and bounded such that $$L_i \le x_i \le U_i\quad \forall i=1,2,\dots,n.$$

How can i define a set of mixed integer linear inequalities such that the feasible solution will be

$$y = \max(x_1,x_2,\dots,x_n).$$

I am not certain of my approach but this is what i have so far

constraint 1: $$y \ge x_i$$;

constraint 2: $$y \ge x_i + (U_i-L_i)w$$;

where $$w\in\{0,1\}$$

Constraint 1 is fine. It imposes $$y \ge \max(x_1,x_2,\dots,x_n)$$. For constraint 2, you need to reverse the inequality and also introduce binary variable $$w_i$$ (with the interpretation that $$w_i=1$$ implies that $$x_i$$ is the maximum): $$y \le x_i + M_i (1 - w_i).$$ We want the constraint to be redundant if $$w_i = 0$$, so take $$M_i = \max(U_1,\dots,U_n) - L_i$$.
Now also impose $$\sum_{i=1}^n w_i=1.$$ (Alternatively, you can relax this to $$\ge 1$$.)
If $$n=2$$, this formulation reduces to the one given here.