Multiple Quantifier Proof

Question

I am completely confused on how to go about proving this multiple quantifier expression.

$$(\forall m\in\mathbb Z)(\exists N\in\mathbb Z)(\forall n\in\mathbb Z)(n\geq N\Rightarrow (n-1)^2 \geq m^2)$$

My translation of this is for every integer $m$, for some integer $N$, and for every integer $n$, if $n$ is greater than or equal to $N$, then $n-1$ squared is greater than or equal to $m$ squared. Does anyone know how to prove this statement? I don't see why it's true because I don't understand how something can be greater than or equal to every integer squared. It doesn't make logical sense.

Any help is greatly appreciated!

Answer 1

Pick $m\in\mathbb Z$, and pick $N\geq |m|+1$. Then if $n\geq N$, $$(n-1)^2\geq(N-1)^2\geq m^2.$$

Answer 2

I don't see why it's true because I don't understand how something can be greater than or equal to every integer squared. It doesn't make logical sense.

It doesn't have to be the same number for every integer, there just has to be at least one example for any integer.   The order of the quantifiers is important.

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Let's look at a less busy statement.

Consider: $\forall m\in\Bbb Z \;\exists n\in\Bbb Z \;(n>m)$

Now that reads: "For any integer there is an integer that is greater than it".   This statement is true.   Pick any $m$, then its successor, $m+1$, is an example of such.

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On the other hand, the way you interpreted the question was more like you were reading the statement:   $\exists n\in\Bbb Z \;\forall m\in\Bbb Z \;(n>m)$

This reads: "There is an integer which is greater than every integer."   This statement is false!   (No integer is greater than itself, for one thing.)

As you see, changing the order of existential and universal quantification can completely change the meaning of the statement.   Be careful.