lean_workbook_plus_31690

|x| + |y| = \left|\left(\frac{x+y}{2}\right) + \left(\frac{x-y}{2}\right) \right| + \left|\left(\frac{x+y}{2}\right) - \left(\frac{x-y}{2}\right) \right| \n \n $\le \left|\left(\frac{x+y}{2}\right) \right| + \left|\left(\frac{x-y}{2}\right) \right| + \left|\left(\frac{x+y}{2}\right) \right| + \left|\left(\frac{x-y}{2}\right) \right|$ $(\because \ |a+b| \le |a| + |b|$ by Triangle Inequality) \n \n $=|x+y| + |x-y|$