Question

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

Answer #1

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

prove that a sequence converges if and only if all subsequences
converge to the same limit

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

Prove that a sequence (un such that n>=1)
absolutely converges if the limit as n approaches infinity of
n2un=L>0

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.

Using the definition of convergence of a sequence, prove that the
sequence converges to the proposed limit.
lim (as n goes to infinity) 1/(n^2) = 0

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

Prove that every bounded sequence has a convergent
subsequence.

show that a sequence of measurable functions (fn)
converges in measure if and only if every subsequence of
(fn) has subsequence that converges in measure

Determine whether the sequence converges or diverges. If it
converges, find the limit. (If an answer does not exist, enter
DNE.)
an = (4^n+1) /
9^n

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