Proving the Union of Intervals is All of N

[fk378]

Homework Statement

Prove that the union of intervals [1,n] from n=1 to n=infinity is all of N.

The Attempt at a Solution

Do I use induction on this? Archimedes? (This question is before the section of Archimedes though). I need help on how to start it!

 

[CompuChip]

Counterexample:
3/2 is in the union, because it is in [1, n] for, for example, n = 2. But 3/2 is not a natural number.

Did you mean: "prove that the union contains N"?

 

[fk378]

I mean that the union of all those intervals from 1 to infinity IS N.

 

[CompuChip]

Since that seems to be false, let's go back a step.
Do you also use the definitions
$$[1, n] = \{ x \in \mathbb R \mid 1 \le x \le n \}$$
(for $n \ge 1$) and
$$N = \{ 1, 2, 3, 4, \ldots \}$$?