REAL ANALYSYS true or false question full answer please If a

REAL ANALYSYS true or false question. full answer please!

If a sequence is bounded and monotone, then it must converge. If a sequence is bounded, then it must contain a convergent subsequence. It is possible to have a convergent sequence that is not Cauchy. It\'s possible to have an open set that is not infinite. If x is a boundary point of A and x NOT AN ELEMENT OF A, then x must be an accumulation point.

Solution

a) true

Every bounded monotone sequence converges

b) true

Every bounded sequence in Rn has a convergent subsequence. ... Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set, because the set is closed.

c) false

every convergent sequence is Cauchy

d) false