6th Swedish Mathematical Society Problems 1966

1.  Let {x} denote the fractional part of x = x - [x]. The sequences x1, x2, x3, ... and y1, y2, y3, ... are such that lim {xn} = lim {yn} = 0. Is it true that lim {xn + yn} = 0? lim {xn - yn} = 0? 2.  a1 + a2 + ... + an = 0, for some k we have aj ≤ 0 for j ≤ k and aj ≥ 0 for j > k. If ai are not all 0, show that a1 + 2a2 + 3a3 + ..