lean_workbook_plus_15622

Let $\hat {x}=(x_1,x_2,.....,x_n)$ and $\hat {y}=(y_1,y_2,.....,y_n)$ where each $x_i$ and $y_i$ is a real number. $(1\leq i\leq n)$ Define $\hat {x}>\hat {y}$ if there exists $k$ such that $1\leq k\leq (n-1)$, $x_1=y_1,x_2=y_2,.........x_k=y_k$, but $x_{k+1}>y_{k+1}$. Show that for $\hat {u}=(u_1,u_2,..........,u_n),\hat {v}=(v_1,v_2,..........,v_n),\hat {p}=(p_1,p_2,..........,p_n)$ and $\hat {q}=(q_1,q_2,..........,q_n)$, if $\hat {u}>\hat {v}$ and $\hat{p}>\hat {q}$, then $(\hat {u}+\hat {p})>(\hat {v}+\hat {q})$.