{xn} is a bounded sequence such that all convergent subsequences converge to the same number

Q: 1 Prove that if a sequence {a„} converges to a, then the sequence {[an]} converges to |a|. Find…

A: Suppose {an} converges to a. Prove that an converges to a. Also, check the converse is true.

Q: Exercise 5. Prove that a non-decreasing (resp. non-increasing) sequence which is not bounded above…

A: Let an be a non decreasing sequence . ⇒an<an+1  ∀n∈ℕ Since an is not bounded above so for any…

Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x*' is point-wise convergent but is not…

A:

Q: B. Prove Bolzano-Weierstrass Theorem. a bounded sequence of real numbers has a convergent…

A: We have to prove

Q: Show that a sequence in R" converges if and only if the i-th coordinates of the sequence converge in…

A:

Q: п — 3 - an

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.