Prove or disprove some statements about linearly independent vectors

Let $A,B,C \in \mathbb{R}^n$ be linearly independent vectors. Prove or disprove each of the following statements.

Proof. Suppose
$x(A+B) + y(B+C) + z(A+C) = O.$

Then we have,

$\begin{align*} && xA + xB + yB + yC + zA + zC &= O \\ \implies && (x+z)A + (x+y)B + (y+z)C &= O \\ \implies && x+z = x+y = y+z = 0 \end{align*}$

by the independence of $A,B,C$ . But then these three equations require $x = y = z = 0$ . Hence, the three given vectors are independent as well $. \qquad \blacksquare$
This is false. Consider
$(1)(A-B) + (1)(B+C) + (-1)(A+C) = A - B + B + C - A - C = O.$

Thus, $x = y = 1$ , $z = -1$ is a nontrivial solution of $x(A-B) + y(B+C) + z(A+C)= O$ , so these vectors are dependent.