Subsequence

Suppose be a sequence and be an increasing sequence. Then is a subsequence of .

Existence of subsequence

Every sequence has a monotonic subsequence.

Proof Hint

• Let be called “good” iff .
• Suppose has infinitely many “good” points. That implies has a decreasing subsequence.
• Suppose has finitely many “good” points. Let is the maximum of those. That implies has a increasing subsequence.

Bolzano-Weierstrass

Every bounded sequence on has a converging subsequence.

Proof Hint

From the above theorem, there is a monotonic subsequence which is also bounded. Bounded monotone sequences converge.

Note

For a set , all statements are equivalent:

• has the completeness property
• is complete
• Bolzano-Weierstrass theorem on

Theorem 1

Suppose is a sequence converging to , and is a subsequence of . Then is converging to .

Proof Hint

Note that .

Theorem 2

Suppose is a sequence diverging to , and is a subsequence of . Then is diverging to .

Proof Hint

Note that .

Subsequence of a cauchy sequence

If is Cauchy and is a subsequence converging to , then converges to .