Question

Detailed proof of the theorem:
Theorem: Every bounded sequence has a limit point.

Answer 1

Since A is bounded, ∃M > 0 / A ⊂ [[M, M]. Cutting the interval in half,

and choose a half that has an infinite number of elements in it (WLOG A1 ≡ [0, M]). Repeat

this process over and over, creating a set of nested intervals, whose widths

tend to zero. Thus there is some p such that $\bigcap$^∞_n=1 An= {p}. Using the intervals created, we

can create an appropriate sequence