Real Analysis Answers

sir please differentiate between applied and pure mathematics also classify the folowing as applied or pure mathematics.Real Analysis,complex analysis,topology,group theory,mathematical statistics,calculus,function alalysis.best regards

a compact space having no finite sub space of (0,1)

tan invers x is uniformly continious or not ?

Suppose f is continuous on all real numbers and f(c) is not equal to zero for some point c in all real numbers. Show that there exists an interval (a, b) containing c such that f is never 0 on this interval

Suppose that the limit as n approaches infinity of Xn=0. If {Yn} is a bounded sequence, prove that the limit as n approaches infinity of XnYn=0

Consider the set of natural numbers N.Is it possible to build a family of subsets of N such that it's cardinality is continuum and for arbitrary two subsets A and B - A includes B or B includes A?

let S be a subset of R and let it bounded below , show that infimum of S exists
[ hint: let A= -s :={ -s: s∈S} , show that A is bounded above so sup A exists . now show that inf S= -sup A ]

show that if a,b element R.then
a. max{a,b}=1/2(a+b+|a-b|) and min{a,b}=1/2(a+b-|a-b|)
b. min{a,b,c}=min{min{a,b},c}

Prove that every infite set is equivalent to one of its proper subsets....

find all x belongs to R that satisfy the eqaction |x+1|+|x-2|=7