Question

If (x_n) is a convergent sequence prove that (x_n) is bounded. That is, show that there exists C>0 such that abs(x_n) is less than or equal to C for all n in naturals

Answer #1

Given that xn is a sequence of real numbers. If (xn) is a
convergent sequence prove that (xn) is bounded. That is, show that
there exists C > 0 such that |xn| less than or equal to C for
all n in N.

Prove that every bounded sequence has a convergent
subsequence.

(a) Prove that the sum of uniformly convergent sequences is also
a uniformly convergent sequence.
(b) Prove that if, in addition to part (a), the sequences are
bounded, then the product is also uniformly convergent.

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.

Prove that, for x ∈ C, when |x| < 1, lim_n→∞ |x_n| = 0.
Note:
To prove this, show that an = xn is monotone decreasing and
bounded from below. Apply the Monotone sequence theorem. Then, use
the algebra of limits, say limn→∞ |xn| = A, to prove that A =
0.

Let (x_n) from(n = 1 to ∞) be a sequence in R. Show that x ∈ R
is an accumulation point of (x_n) from (n=1 to ∞) if and only if,
for each ϵ > 0, there are infinitely many n ∈ N
such that |x_n − x| < ϵ

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

Problem 1 Let {an} be a decreasing and bounded
sequence. Prove that limn→∞ an exists and
equals inf{an}.

Prove or disprove that if (xn) is an unbounded sequence in R,
then there exists n0 belongs to N so that xn is greater than 10^7
for all n greater than or equal to n0

Prove that if a sequence an is NOT bounded above,
then limsup an = infinity.

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