To show that the union of the intervals [3 - 1/n, 3 + 1/n] for n ∈ N is not closed, we need to find a limit point of the union that is not contained within the union itself.
Consider the point x = 3. This point is a limit point of the intervals [3 - 1/n, 3 + 1/n] because for any ε > 0, we can always find an interval in the union that contains x within the interval (3 - ε, 3 + ε). This is because as n approaches infinity, the intervals shrink and eventually contain the point x = 3.
However, x = 3 does not belong to any interval in the union [3 - 1/n, 3 + 1/n] for n ∈ N. In other words, x is not an element of the union itself.
Therefore, we have found a limit point (x = 3) that is not contained within the union [3 - 1/n, 3 + 1/n] for n ∈ N, which means the union is not closed.