No idea how to do this one. I know that if ($b_n$) is convergent it's bounded and either increasing or decreasing, hence ($a_n$) is going to converge but I have no idea how to prove that or find the limit.

Given two sequences of positive real numbers ($a_n$), ($b_n$) such that n belongs to the naturals. ($b_n$) is convergent and ($a_1 + a_2 + ... + a_n$) is less than or equal to $b_n$ for every positive integer $n$. Prove that ($a_n$) must converge and what is its limit?

  • $\begingroup$ Natural numbers as in (1, 2, 3, 4,...) not including $\endgroup$ – dahaka5 Nov 26 '15 at 17:25

Let $c_n \equiv a_1 + a_2 + ... + a_n$. Since you are given that $0\leq c_n\leq b_n$ and that $b_n$ is convergent, it follows that $c_n$ must converge as well, which implies that

$$\lim_{n\to\infty}c_n = \sum_{j=1}^\infty a_n$$

is finite. This can only be the case if $$\lim_{n\to\infty}a_n=0.$$


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