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(a) For each n € N, the interval,3-. is closed in R. E Show that Un U-1,3- n=1 ] is not closed

User Farray
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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.
User Remi Collet
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