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1. ### Use Extreme Value Theorem

Suppose f:[0,1]->R is continuous, f(0)>0, f(1)=0. Prove that there is a X0 in (0,1] such that f(Xo)=0 & f(X) >0 for 0<=X<Xo (there is a smallest point in the interval [0,1] which f attains 0) Since f is continuous, then there exist a sequence Xn converges to X0, and f(Xn) converges to f(Xo)...
2. ### Prove G contains a cycle of length at least k+1

This is a graph theory related question. Let G be a simple graph with min. degree k, where k>=2. Prove that G contains a cycle of length at least k+1. Am I suppose to use induction to prove G has a path length at least k first, then try to prove that G has a cycle of length at least k+1...
3. ### How to prove a simple graph is 2-connected?

Problem: "Let G be a simple graph on n vertices such that deg(v)>= n/2 for every vertex v in G. Prove that deleting any vertex of G results in a connected graph." Well, I tried to find the min. case. Let k be the min. deg. of vertex in a simple graph, n is number of vertices in G so k =...
4. ### Graph Theory - bipartite related proof

How to prove that the number of edges in a simple bipartite graph with n vertices is at most n^2/4? Definition of bipartite graph: a graph whose vertex-set can be partitioned into two subsets such that every edge has one endpoint in one part and one endpoint in the other part. I try to...
5. ### Convergence of Sequence Does {An^2} converges => {An} converges? How to prove it?

Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove it?" Does sequence {An^2} converges implies to sequence {An} converges? True or False. How to prove it? I kinda think it is false, but couldn’t think of any counterexample to directly proof it. So I try to use...
6. ### Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

How to prove the open intervals (0,1) and (0,2) have the same cardinalities? |(0, 1)| = |(0, 2)| Let a, b be real numbers, where a<b. Prove that |(0, 1)| = |(a, b)| ----------------------------------- |(0,1)| = |R| = c by Theorem ----------------------------------- I know that we...
7. ### Set Theory: Prove the set of complex numbers is uncountable

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...
8. ### Prove set S is countable iff there exists a surjective/injective function

(a) A nonempty set S is countable if and only if there exists surjective function f:N->S (b) A nonempty set S is countable if and only if there exists a injective function g:S->N There are two way proves for both (a) and (b) (a-1) prove if a nonempty set S is countable, then there exists...
9. ### Cardinality Problem: Prove |A| < |N|

Prove cardinality of every finite nonempty set A is less then cardinality of natural number N |A|<|N| set A is nonempty finite set natural number N is denumerable (infinite countable set) |A|<|N| if there exist a injective (one-to-one) function f: A->N, but NO bijective function, which...