Can Trivial Conditions Validate an Implication in Mathematical Proofs?

[knowLittle]

Homework Statement

Let $n\in N$. Prove that if $|n-1|+|n+1|\leq 1,$ then $|n^{2}-1|\leq 4$

Homework Equations

The Attempt at a Solution

I am trying to show by a counter example that this statement is not true.
Consider this statement:
$|n-1|+|n+1| \leq 1$
Assume :
|n-1|=n-1 >0 and |n+1|=n+1>0
Then, by our former statement, $n-1+n+1 \leq 1,$
which gives
$2n \leq 1 , \text{where n} \in N$
Now, divide by 2, and $n \leq \dfrac{1}{2}$, which is not possible since $n \in N$

Call $|n-1|+|n+1| \leq 1$ , P.
In logic, P→Q. If P is false, then Q does not matter, the implication is always true.

Is this non-elegant proof correct?

 

[HallsofIvy]

You said you were "trying to prove by a counter example that this statement is not true." First, this is not a counter example. A "counter example" would be a specific n, satisfying the hypothesis, such that the conclusion is not true.

Further, you said yourself that "In logic, P→Q. If P is false, then Q does not matter, the implication is always true." So, by showing that there is NO n in N such that $$|n- 1|+ |n+1|\le 1$$, you have shown that the hypothesis is always false and so have shown that the statement itself is true.