Example of an unbounded sequence whose convergent subsequences converge to same limit

Question

"A bounded sequence of real numbers converges to x if every convergent subsequence of the sequence converges to x." I require a counterexample to prove that the theorem fails if the hypothesis that the sequence is bounded is dropped. Also, am I correctly interpreting the question?

Answer 1

Consider the trivial example $(x_n)_{n\in\mathbb{N}}$ given by $x_n=n$, which obviously does not converge. This sequence has no convergent subsequence, so the condition from the theorem holds vacuously for any $x$ (i.e. for any $x\in\mathbb{R}$, any convergent subsequence of $(x_n)$ converges to $x$).

Answer 2

Consider the sequence $(x_n) \in {}^{\def\N{\mathbf N}\N}\def\R{\mathbf R}\R$ given by $$ x_n = \begin{cases} n & \text{if $n$ is even}\\ 0 & \text{if $n$ is odd} \end{cases} $$ Then every convergent subsequence of $(x_n)$ converges to 0, but $(x_n)$ does not converge.

Answer 3

By definition, every unbounded sequence contains a sub-sequence that diverges to plus or minus infinity.

Thus not all sub-sequences can converge to the same value.