I was proving a problem which states that: let $\{x_n\}_n$ be a monotone sequence of real numbers. Show that $\{x_n\}_n$ is convergent if and only if it is bounded.
I have proven that if the sequence is monotone and bounded, it is convergent. But must I prove the reverse case where I assume that the sequence is monotone and convergent and prove boundedness? I assume this is trivial because if the sequence is increasing or decreasing and is bounded, then the limit must be the supremum or the infimum.
Best Answer
It's trivial, but for a different reason: a convergent sequence in any metric space is bounded. Suppose $\{x_n\}$ isn't bounded. Then no open ball centered at $x$ contains all points of the sequence. So for each $k$, we can find a point $x_{n_k}$ such that $d(x,x_{n_k})>k$. Take $\varepsilon=1$. Then for any $N$, we have a positive integer $n_k$ such that $d(x,x_{n_k})>k>1$. Hence, $x_n$ does not converge to $x$, a contradiction.
In fact, any Cauchy sequence is bounded. It's a good exercise to prove that as well.