Real Analysis Answers

If fn converges to f and FN is bounded on a set S prove that {fn} is uniformly bounded on S

Determine which of the following functions are of bounded variation on[0,1]

a. f(x)= x^2sin(1/X) if x#0 ,f(0)=0

b. f(x)= √x sinx,if x#0,f(0)=0

If f:R^2 to R defined by f(X,y) ={x^2y^3/x^4+y^2. if (x,y)#(0,0)

0 if (X,y) =(0,0). Find the directional derivative of f(X,y) at (0,0)?

Prove that an open interval in R is an open set and a closed intervals is a closed set

Find limit of sequence =n+2/3-n

If f(X,y)={xy/x2+y2, if(X,y)#(0,0)


0 if (X,y)=(0,0)

Check the continuity CF F(X,y) at (0,0)

Prove that lim(Xn) =0 iff lim(|Xn|)=0 . Give an example to show that convergence of (|Xn|) need not imply the convergence of (Xn)

With the help of sequences, calculate √5 correct upto 5 decimals.

Show that if X and Y are sequences such that X and X + Y are convergent, then Y is convergent.

Use the order completeness property to show that the set S= {n/n+7:n belong to N} has a supremum and infimum