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 $\displaystyle\lim_{n\to +\infty}a_n = 0$?


Thank you in advance!

  • 1
    $\begingroup$ Hint: show your efforts. See How to ask a good question. $\endgroup$ – Saad Apr 22 at 12:37
  • $\begingroup$ $a_n$ must be non-negative, unless we can have counterexamples like $$b_n={1\over n}$$ and $$a_n=-1$$ $\endgroup$ – 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.

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