Question

A sequence X = (an),an = sin(n) is​

Answer 1

Answer:

sin (n) is dense in [-1,1] and so this obviously means that there are so many convergent subsequences for this sequence. We can construct an example

as follows:

for a positive integer "n" there exist an integer sequences "m_k" and "l_k" which "2(l_k)pi+m_k" is convergent to "n" (this is implied by the fact that if "a" is irrational number, then the set of "ma+n" where m,n are integer, is dense in "R").

Now by continuity sine function we have that

Sin(2(l_k)pi+m_k) goes into Sin(n) which can be concluded that

Sin(m_k) goes into Sin(n) as we wish.