I am given a sequence $\{a_n\}$ with $a_n > 0$ and $b_n = a_n + 1/a_n$.
I am first asked to assume that $a_n \ge 1$ and show that the convergence of $\{b_n\}$ implies the convergence of $\{a_n\}$.
My argument is:
Since $\{b_n\}$ converges, $\{b_n\}$ is Cauchy, hence for every $\epsilon > 0$ some $N$ such that for $n_i,n_j > N$, $|b_{n_i} - b_{n_j}| < \epsilon$. Then $|b_{n_i} - b_{n_j}| = |a_{n_i} - a_{n_j} + 1/a_{n_i} - 1/a_{n_j}| < \epsilon$. Suppose that $a_{n_i} = a_{n_j}$. Then clearly $|a_{n_i} - a_{n_j}| < \epsilon$. Alternatively, suppose without loss of generality that $a_{n_i} > a_{n_j}$. Then $1/a_{n_i} - 1/a_{n_j} < 0$ while $a_{n_i} - a_{n_j} > 0$. It is straightforward to observe that $-(1/a_{n_i} - 1/a_{n_j}) \le a_{n_i} - a_{n_j}$ because of the condition $a_n \ge 0$, so it follows that $0 < |a_{n_i} - a_{n_j}| < |a_{n_i} - a_{n_j}|$, so then $|a_{n_i} - a_{n_j}| < |a_{n_i} - a_{n_j} + 1/a_{n_i} - 1/a_{n_j}| < \epsilon$, hence $\{a_n\}$ is Cauchy and therefore converges.
However, I am next asked to assume that $\{b_n\}$ converges but only that $a_n > 0$ and demonstrate, through construction of a counterexample, that it does not necessarily follow that $\{a_n\}$ converges. I am not entirely sure how to do this.