Problem solving seminar VII
25. Let f : [0, ∞) → R be a function satisfying f(1) = 1 and
f (x) =
1
x2 + f(x)
.
Prove that
lim
x→∞
f(x)
exists and is less than 1 + π
4
.
26. Let G be the abelian group defined by generators x, y and z, and relations
15x + 3y = 0
3x + 7y + 4z = 0
18x + 14y + 8z = 0
(1) Express G as a direct product of two cyclic groups. (A direct product is the
same as a direct sum.)
(2) Express G as a direct product of cyclic groups of prime power.
(3) How many elements of G have order two?
27. Let X be a metric space and let V be a finite-dimensional subspace of the
vector space of continuous real valued functions on X. Prove that there is a basis
{f1, f2, . . . , fn} for V and points x1, x2, . . . , xn in X such that
fi(xj) =
1 for i = j,
0 for i = j.
1