数学1

1. Heine-Borel Theorem

compact-->bounded: finite open cover B (xi,1); pick r=max {||x1-xi||}+2

compact-->closed: consider B(y, r); 交open cover (finite) 为空集: r=min{||xi-y||/3}

closed & bounded-->compact:

-consider set A: {x, [a,x] can be covered by finitely many Ui}

least upper bound c is in A

-b=c

(Three Hard Theorems)

 

2. Prove that if a subset A in Rn is compact, then any subsequence of points in A has a convergent subsequence.

Hints: 

1D space: Balzano-Weierstrass Theorem

Define a projection function f: A->R; f continuous (preimage) 

Run the algorithm

 

标签：compact,xi,consider,--,数学,Theorem,open
来源： https://www.cnblogs.com/EstherZhao/p/16209342.html