Is the Interval [a,b] Sequentially Compact?

[math8]

I'd like to show that [a,b] is sequentially compact. So I pick a sequence in [a,b] , say (xn).

case 1:range(xn) is finite
Then one term, say c is repeated infinitely often. Now we choose the subsequence that has infinitely many similar terms c. It converges to c.

case 2: range of (xn) is infinite.
I know I can prove this using the Bolzano Weierstrass theorem that says that every infinite bounded subset of R has a limit point. But in this case I'd have to prove the theorem. How?
Or I may Try and subdivide [a,b] in two parts and say that one part must contain infinitely many points of the range, and pick a term x_n1 in that part. Next I can choose x_n2 since I have infinitely many terms to choose from. And then continue to subdivide... But then I'm stuck. I am not sure what I need to do from there or how exactly I need to write this down (how the subdivision part works exactly).
Thanks for your help.

 

[morphism]

The subdivision part works like this. Say you divide [a,b] into two closed intervals I_1 and J_1 (for instance, I_1 = [a,c] and J_1 = [c,b]), where say I_1 contains infinitely many terms of the sequence. Then further split I_1 into I_2 and J_2 where I_2 contains infinitely many terms of the sequence. Proceed like this to get a decreasing sequence of closed intervals I_1 >= I_2 >= ... of decreasing 'length'.

Show that the sequence {x_n_i} you formed is Cauchy.