prove that {1/n} n=1 to inf is not compact in R with usual metric. suppose use the adjoint point {0} to {1/n} n=1 to inf

To prove that the set {1/n} for n=1 to infinity is not compact in R with the usual metric, we can use the concept of open covers and the property that a set is compact if and only if every open cover has a finite subcover.

Step 1: Define the set A = {1/n} for n=1 to infinity and include the point 0. So, A = {0, 1, 1/2, 1/3, ..., 1/n, ...}.

Step 2: Define an open cover of A. An open cover of A is a collection of open sets such that A is a subset of the union of these open sets. We can define an open cover C of A as follows: C = {(1/(n+2), 1/(n-1)) for n=2 to infinity} union (1/2, infinity). You can verify that every point in A is contained in at least one of these open intervals, so C is indeed an open cover of A.

Step 3: Show that C does not have a finite subcover. A finite subcover of C is a finite collection of sets from C that still covers A. However, no matter how many sets you remove from C, you will always be left with the point 0 not covered, because the interval containing 0, (1/2, infinity), is not part of any finite subcollection of C.

Therefore, the set A = {1/n} for n=1 to infinity including the point 0 is not compact in R with the usual metric, because it has an open cover that does not have a finite subcover.