Limits question - How to solve it

Question

Let $a_n$ be a bounded sequence of strictly positive real numbers for which the limit of $a_n + \frac{1}{a_n}$ is $+\infty$ as $n\to\infty$. Prove that the limit of $a_n$ as $n\to\infty$ is $0$

I dont know where to start with this one so any help is appreciated

Answer 1

Hint: Suppose that you don't have $\lim_{n\to\infty}a_n=0$. Then, for some $\varepsilon>0$, the inequality $|a_n|\geqslant\varepsilon$ holds infinitely many times. Since the $a_n$'s are strictly positive, this means that $a_n\geqslant\varepsilon$. What does this, together with the boundness of $(a_n)_{n\in\mathbb N}$, tells you about the sequence $\left(a_n+\frac1{a_n}\right)_{n\in\mathbb N}$?

Answer 2

Assume that $(a_n)$ does not tend to $0$ if $n \to \infty$. Since $(a_n)$ is bounded, there is an accumulation point $a \ne 0$ of $(a_n)$. Then we get a subsequence $(a_{n_k})$ with $a_{n_k} \to a$ as $k \to \infty$. Therefore:

$a_{n_k}+\frac{1}{a_{n_k}} \to a+1/a$ as $k \to \infty$.

This is a contradiction , since $a_{n_k}+\frac{1}{a_{n_k}} \to \infty$ as $k \to \infty$.