If (an) is a sequence satisfying しに! Prove that If (an) is a sequence satisfying しに! Prove that
Prove that \に1 に1
Prove by mathematical induction that for the logistic map with , the sequence can be expressed as . r.r( 1-2) 7 f2ok (ro) に! 2 2 ok (xo r.r( 1-2) 7 f2ok (ro) に! 2 2 ok (xo
3) Let (an)2- be a sequence of real numbers such that lim inf lanl 0. Prove that there exists a subsequence (mi)2-1 such that Σ . an, converges に1
tuen Lim nwm し,m im Prove thart <ly 2 〉 2.
2. 15 pts] Suppose E,, E. , En are independent events. Prove that に!
(c) A sequence {2n} satisfying 0 < In < 1/n where E(-1)"In diverges.
Exercise 1.5. Prove that if A and B are sets satisfying the property that A \ B = B \ A, then it must be the case that A = B. Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, A4B = (A ∪ B) \ (A ∩ B). Exercise 1.7. Verify the second assertion of Theorem 1.3.4, that for any collection of sets {Ai}i∈I, \ i∈I Ai !c = [ i∈I...
Show that the sequence (e2ni(mx-+ ry)y is an orthonormal set in し2 ([0, 1] × [0, 1]). Here the L2 norm is 1/2 げ(x,y)?dxdy, Jo Jo Show that the sequence (e2ni(mx-+ ry)y is an orthonormal set in し2 ([0, 1] × [0, 1]). Here the L2 norm is 1/2 げ(x,y)?dxdy, Jo Jo
Prove this proposition please. 4.2.19) Proposition. Whenever a <b, there is a smooth function f satisfying f(2)= { € (0,1), a<:<b, VIVA *> 6. For obvious reasons, such a function is called a bump function. o M Figure 61. A bump function
Problem 3. Prove or give a counter example 1. If an converges to a real limit then limn700 (m)" = 0. 2. If an is a positive sequence satisfying limn+ ()" = 0 then it con- verges.