Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit

Theorem
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence.

Let $x \in \R$.

Suppose that:


 * every subsequence $\sequence {x_{n_j} }_{j \in \N}$ of $\sequence {x_n}_{n \in \mathop \N}$ has a subsequence $\sequence {x_{n_{j_k} } }_{k \in \N}$ such that:


 * $x_{n_{j_k} } \to x$

Then:


 * $x_n \to x$

Proof
, suppose that:


 * $x_n$ does not converge to $x$.

Then, there exists some $\epsilon > 0$ such that for every $k \in \N$ there exists $n_k \ge k$ such that:


 * $\size {x_{n_k} - x} \ge \epsilon$

Let $\sequence {x_{n_{j_k} } }_{k \in \N}$ be a subsequence of $\sequence {x_{n_j} }_{j \in \N}$.

Then:


 * $\size {x_{n_{j_k} } - x} \ge \epsilon$

for each $k \in \N$.

In particular, there exists no $N \in \N$ such that:


 * $\size {x_{n_{j_k} } - x} < \epsilon$

for each $k \ge N$, so:


 * $\sequence {x_{n_{j_k} } }_{k \mathop \in \N}$ does not converge to $x$.

So:


 * $\sequence {x_{n_j} }_{j \mathop \in \N}$ is a subsequence of $\sequence {x_n}_{n \mathop \in \N}$, but has a subsequence that does not converge to $x$.

This contradicts our hypothesis on the subsequences of $\sequence {x_n}_{n \mathop \in \N}$.

So $x_n$ converges to $x$.