Limit superior and inferior of a sequence that satisfies the asymptotic formula $\sum\limits_{n\leq x} a_n \sim x $

Question

Suppose $\{a_n\}$ is a sequence such that $\sum\limits_{n\leq x} a_n \sim x $. I have to show that: $$ \liminf a_n\leq 1 \leq \limsup a_n$$

I've no idea on how to approach this, in all honesty.

Answer 1

Hint: Suppose it is not true and $\lim\inf a_{n}=\delta>1$. Then, for large enough $n$, $a_{n}\geq \frac{1+\delta}{2}>1$ for all $n$ (since the inf of the remaining terms converges up to $\delta$)... once all your terms are strictly bigger than one, the rest of the sum must be growing faster than $x$.