1. The statement "Since $n^{1/2} \geq 0$ for real $n$" means the square root of any real number $n$ is always non-negative. 2. The inequality "$-2c \geq 0$" implies that multiplying $c$ by $-2$ results in a non-negative number. 3. To solve for $c$, divide both sides of the inequality by $-2$. Remember, dividing by a negative number reverses the inequality sign: $$-2c \geq 0 \implies c \leq 0$$ 4. Therefore, the conclusion "$c \leq 0$" follows logically from the inequality. 5. If step 5 in your original problem states "$c \leq 0$" based on the above reasoning, then yes, this is the same as the given statement. In summary, the reasoning is correct and the statements are equivalent.