Question

Prove: If x is a sequence of real numbers that converges to L, then any subsequence of x converges to L.

Answer #1

Prove that if a sequence converges to a limit x then very
subsequence converges to x.

Suppose (an) is an increasing sequence of real numbers. Show, if
(an) has a bounded subsequence, then (an) converges; and (an)
diverges to infinity if and only if (an) has an unbounded
subsequence.

Suppose {xn} is a sequence of real numbers that converges to
+infinity, and suppose that {bn} is a sequence of real numbers that
converges. Prove that {xn+bn} converges to +infinity.

Prove that X is totally bounded if every sequence of X has a
convergent subsequence. Please directly prove it without using any
theorem on totally boundedness.

Show that sequence {sn} converges if it is monotone
and has a convergent subsequence.

Give a sequence of rational numbers that converges to √5 (i.e.
converges to L where L^2=5). No proof needed.

Suppose (an), a sequence in a metric space X, converges to L ∈
X. Show, if σ : N → N is one-one, then the sequence (bn = aσ(n))n
also converges to L.

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Prove that if a sequence is a Cauchy sequence, then it
converges.

Prove that every bounded sequence has a convergent
subsequence.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 21 minutes ago

asked 21 minutes ago

asked 33 minutes ago

asked 48 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago