Question: How to show that the sequence $n\cos(\frac{1}{n})$ has no convergent subsequence or in general how to show that sequence has no convergent subsequence.
My attempt: If the sequence is bounded then it has a convergent subsequence by Bolzano Weierstrass theorem but, when sequence is not bounded then it may or may not have convergent subsequence! For example: The sequence $1,2,3,1,4,5,1,6,7,1,…$ is Not bounded but has convergent subsequence, which is the constant sequence $1,1,1,…$
But here, in the case of the sequence given in the question,I am unable to determine it is bounded or not and it has convergent subsequence or Not! Please help me…
Best Answer
Notice that $n\cos(1/n) > n/2$ for all $n$ large enough. Can you take it from here?