In: Advanced Math

Real Analysis: Prove a subset of the Reals is compact if and only
if it is closed and bounded. In other words, the set of reals
satisfies the Heine-Borel property.

1.
Prove that any compact subset ot a metric space is closed and
bounded.
2. Prove that a closed subset of a compact set is a compact
set.

Prove Mn(reals) is a group under matrix addition. Note,
Mn(reals={A|A is a real nxn matrix}) Please show all steps and do
not write in script!

Prove that a subspace of R is compact if and only if it is closed and bounded.

Prove that the product of a finite number of compact spaces is
compact.

Prove that the union of a finite collection of compact subsets
is compact

Please prove that: A nonempty compact set S of real numbers has
a largest element (called the maximum) and a smallest element
(called the minimum).
By the way, I think a minimum is provided by -max(-S)

If X is any topological space, a subset A ⊆ X is compact (in the
subspace topology) if and only if every cover of A by open subsets
of X has a finite subcover.

Question:
Prove that the intersection of two compact sets is compact, using criterion (2).
Prove that the intersection of two compact sets is compact, using criterion (1).
Prove that the intersection of two compact sets is compact, using criterion (3).
Probably the most important new idea you'll encounter in real analysis is
the concept of compactness. It's the compactness of [a, b] that makes a
continuous function reach its maximum and that makes the Riemann in-
tegral exist. For...

Prove or disprove if B is a proper subset of A and
there is a bijection from A to B then A is infinite

Suppose that x is real number. Prove that x+1/x =2 if and only
if x=1.
Prove that there does not exist a smallest positive real number.
Is the result still true if we replace ”real number” with
”integer”?
Suppose that x is a real number. Use either proof by
contrapositive or proof by contradiction to show that x3 + 5x = 0
implies that x = 0.

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