Real Analysis Answers

Prove that if f and g are Riemann integrable on [a, b], then f · g and f + g are Riemann integrable on [a, b].

Prove that a subset of a set of measure zero has measure zero.

Show that the union of two sets, each of measure zero, has measure zero

Let fn(x)= nx/(1+nx) is not uniformly convergent on [0,1]

Let fn(x)= x^n is not uniformly continuous on [0,1] but is uniformly continuous on [0,k]

Check the whether the set { 2/7, 2/8, 2/9,..} is countable or not. Also a give example of proper subset of R which is uncountable

Check whether the function, ,f defined below, is uniformly continuous or not:

f(x)=x^(1/2), x∈[1,2]

Prove or disprove the following statement

‘ Every strictly increasing onto function is invertible'

Is every onto strictly decreasing function invertible? Justify your answer.

All strictly monotonically decreasing sequences are convergent.

True or false with full explanation