Real Analysis Answers

Show that the functions f (x, y) = ln x - ln y and g(x, y) =(x2+2y2)/2xy are functionally dependent.

Check whether the set of integers is countable or not.

Let {bn} be a sequence such that bn = 1 + (1/n). Prove that inf(bn) = 1

using delta - epsilon definition prove that lim(x-->0) (sinx/x) = 1

Prove shortly that |d(x,z)-d(y,z)|≤d(x,y) Give its Hint or prove it shortly.

Q. Is d(x, y) =√(│x-y│) a metric space?(solve it)

Evaluate ∫∫∫z2dx dydz
S
,where S is the solid region between the spheres ρ =1 and
ρ = 2, by using spherical coordinates.

true/false? prove..
Every function differentiable on [a, b] is bounded on [a, b]

If a ∈ R is such that o , ≤ a ≤ ε ∀ ε > 0 ,then show that a = 0.

Show that the sequence (f) where f(x)= x/(1+2nx^2), x∈ [1,∞[ is uniformaly convergent in [1,∞[