Could someone please check my work

Suppose That X,, X2, ... . In are real numbers. Prove That ( X, + x 2 + ... + Xn | = / X,1 + / x2/ + ... + / xnjusing induction on n. " Generalized THangle Base Case : Inequality Ler PCni) be The statement { a; ) = = lail , for all neIN Then PCI) asserts That LHS = ( & a; ) = Jail E Jail = 1 a , 1 = RHS . LHS = 1Q, 1 = RHS Thus, PCI) is True, which establishes The basis for induction. Inductive step Assume That " MT a ; = sail is True for some KEIN. K+ 1 K + 1 WTS Q ; = ist K+ 1 LHS = & a; 1= 1 K ait axtl Sail+ akti by Theorem 3. 2.10 (2) for The Triangle inequality Use The assumption ) = EJail + art 2 Jail = Rt'S By The 3.1.2, Jail, for all n E IN