Real Analysis Answers

Using the ε −δ definition, show that lim 3( x )5 1

Show that T1-space X is regular iff for each point' a' element of X and each open set U containing 'a' ,there is an open set W containing 'a' whose closure is contained in U

Prove or disprove

a. Every topological space is metrizible

b. Any metric defined on X(#0) induces a topology on x

Give an example of a regular space that is not normal

Show that the set of rational numbers with the subspace topology of R is disconnected

Prove that Hilbert space is seperable

Let f element R(alpha) on [a,b] where alpha is of bounded variation on [a,b] and let v(x) denote the total variation of f on [a,x] for each x in [a,b] and let v(a) =0 , show that | integral a to b f d alpha |less than or equal to integral a to b |f| dv less than or equal to M.v(b) where M is an upper bound for |f| on [a,b]

Use Euler summation formula,or integration by parts in a Reimann stieltjies to show that

Summation k from one to infinity 1/k= log n- integral one to n x-[x]/x^2 dx+1

let {an} be a decreasing sequence of positive terms .prove that the series summation an sin (nX) converges uniformily on R if and only if nan tends to 0 as n tends to infinity

If f(x,y) = {1 if (x,y) = (0,0)

sin(x,y) if (x,y)#(0,0).find lim(x,y) converges to (0,0) f(x,y)