This question is based on an exercise in real analysis.
Given $a_n \geq 0$ for every $n\in \mathbb{N}$. Suppose the limit of $b_n = (-1)^na_n$ exists. Prove that $a_n$ converges to zero.
THOUGHTS:
By definition of limits, I need to show that given any $\epsilon>0$, there exists $N$ such that for all $n>N$,
$$
|a_n-0|<\epsilon.
$$
But I do not have any explicit expressions for the two sequences $a_n$ and $b_n$. How can I prove the convergence?
Best Answer
If a sequence converges, every sub sequences converge, and converge toward the same limit.
In particular $(d_n=b_{2n})$ and $(e_n=b_{2n+1})$ converge to the same limit l. You should be able to show that $l = 0$.