MathDB

2

Part of 2012 Pre-Preparation Course Examination

Problems(2)

convergent sequences

Source: Iran PPCE 2012-Analysis exam-P2

2/14/2012
Suppose that lim⁡n→∞an=a\lim_{n\to \infty} a_n=a and lim⁡n→∞bn=b\lim_{n\to \infty} b_n=b. Prove that
lim⁡n→∞1n(a1bn+a2bn−1+...+anb1)=ab\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab.
limitreal analysisreal analysis unsolved
Number of subspaces not less than cardinality of the field

Source: Iran PPCE 2012-Linear Algebra exam-P2

2/16/2012
Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.
vectorlinear algebralinear algebra unsolved