Suppose that x1, x2, x3, ... ,xn are real numbers. Prove (using mathematical induction) that
l x1 + x2 + ... + xn l < lx1l + lx2l + ... + lxnl
You will prove this for $\displaystyle n\geq 2$. If this statement is true for $\displaystyle k$ variables i.e. $\displaystyle |x_1+...+x_k| \leq |x_1| + ... + |x_k|$ we shall prove it is true for $\displaystyle k+1$ variables. In the expression $\displaystyle |x_1+...+x_k+x_{k+1}|$ think of it as $\displaystyle |(x_1+...+x_k)+x_{k+1}|$ but this is less than or equal to $\displaystyle |x_1+...+x_k| + |x_{k+1}|$ but this is less than or equal to $\displaystyle |x_1|+...+|x_k|+|x_{k+1}|$.