Boltzano-Weierstrass Theorem

Let be a Bounded real sequence.
Then contains a convergent subsequence.

Proof

By considering half intervals.
Alternatively, can use increasing bounded sequence converges property.

sequences

Theorem

Same statement as for Real sequences

Proof

To prove it, first find a subsequence such that the first coordinate converges.
Then find the subsequence of that subsequence s.t. the second coordinate converges…