Convergence of Limsup and Liminf

Theorem
Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $\mathbb{R}$.

Let the limit superior of $$\left \langle {x_n} \right \rangle$$ be $$\overline l$$.

Let the limit inferior of $$\left \langle {x_n} \right \rangle$$ be $$\underline l$$.

Then $$\left \langle {x_n} \right \rangle$$ converges to a limit $$l$$ iff $$\overline l = \underline l = l$$.

Hence a bounded sequence converges iff all its convergent subsequences have the same limit.

Proof

 * First, suppose that $$\overline l = \underline l = l$$.

Let $$\epsilon > 0$$.

By Terms of Bounded Sequence Within Bounds $$\exists N_1: \forall n > N_1: x_n < l + \epsilon$$.

Similarly, $$\exists N_2: \forall n > N_2: x_n > l - \epsilon$$.

So take $$N = \max \left\{{N_1, N_2}\right\}$$.

If $$n > N$$, both the above inequalities hold at the same time.

So $$l - \epsilon < x_n < l + \epsilon$$ and so by Negative of Absolute Value $$\left|{x_n - l}\right| < \epsilon$$.

Thus $$x_n \to l$$ as $$n \to \infty$$.


 * Now suppose that $$\left \langle {x_n} \right \rangle$$ converges to a limit $$l$$.

Then by Limit of a Subsequence, all subsequences have a limit of $$l$$ and the result follows.