# Sequences convergence property

Let $$(a_n)_{n\in\mathbb{N}}$$ and $$(b_n)_{n\in\mathbb{N}}$$ be two real-valued sequences. Suppose that $$a_n\leq b_n$$ for all $$n\in\mathbb{N}$$ and $$\lim_{n\to +\infty}b_n = 0.$$

I would like to know: it is true that also $$\displaystyle\lim_{n\to +\infty}a_n = 0$$?

Why?

• $a_n$ must be non-negative, unless we can have counterexamples like $$b_n={1\over n}$$ and $$a_n=-1$$ – Mostafa Ayaz Apr 22 at 12:40
Not if we can take $$a_n$$ to be negative. Suppose $$b_n =0, a_n = -1$$ for all $$n$$. Then $$a_n \leq b_n, \lim_n b_n = 0,\lim_n a_n = -1 \neq 0$$. If $$0 \leq a_n \leq b_n$$, then yes, as a consequence of the Squeeze Theorem.