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Mathematics & Economics
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Math Problem
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Proving the Real Variable Theory

Math Problem Instructions:

1.a)Prove that (0, 1) and [0,1) are not compact by constructing an open cover for each that has no finite subcover. (Short – you need only produce the cover, not show it works.). Similarly show that the set [0,1) union (1,2] is not compact.
b)Given S, an arbitrary set of real numbers, and t, an arbitrary limit point of S with t is not in S, construct an open cover of S with no finite subcover. Use the same idea as b, treating “t” in the
same way you dealt with the number 1. This time, do prove that no finite subcover exists.
c) From this, prove that if F is compact, then F must be closed.
d) Prove that if F is compact, then F must be bounded.

Math Problem Sample Content Preview:

Real Variable Theory
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Real Variable Theory
Question 1a
The main reason the interval is never compact is because it lacks the Heine-borel characteristic. This means that there is a open cover (0,1) that can never be reduced finite subcover.
Un = (an, 2), where an 0
We must define that In = (1/n, 2), which implies that even when we include many In, the interval cannot be covered. This is due to the finite cover that existed. Therefore, the (0,1) and (0,1) are never compact.
When we follow the same method, we can prove that (0,1) and (1,2) is not compact.
Question 1b
Since S is a set of real numbers, t is compact if and only if every individual open...
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