Real Analysis Answers

Define the functions of bounded variation. Show that if f is continuous on [a,b], and if 𝑓′ exists and is bounded in the interior, say  𝑓′(𝑥) ≤ 𝐴 for all 𝑥 in  𝑎,𝑏 ,then f is of bounded variation on [a,b].

If f is monotonic on [a,b], then show that f is of bounded variation on [a,b]. 

Show that the series , ∞Σ n=1 (x/1+n^2x^2) is uniformly convergent in [ k,1] for any k>0.

Show that the series , ∞Σ n=1 (x/1+n^2x^2) is uniformly convergent in [ k,1] for any k>0.

Give an example of an infinite set with finite number of limit points, with proper justification.

Find the truth value of two or more quantifiers

Show that f is continuous on a,b and if f' exists is bounded on interior, says f'x less than A for all x in a,b then the f is of bounded variation

If y is not zero and x, y belongs to Q then prove that x/y belongs to Q

 If S := {1/n - 1/m: n, mEN}, find inf S and sup S