2. (*) Suppose f : R a R is a function satisfying 33... W) : .331... ft\") : -00. (See Homework 4 #11 for the denitions of limits of this type.) (a) Show that if (3211) is any sequence of real numbers subh that (f(a:n)) either converges to a real number or diverges to +00, then ($71) is bounded. (b) Prove that if such a function f is continuous everywhere on R, then f is bounded above. Hint: Show rst that iff is not bounded above, then there exists a sequence (In) with at\") +00. Does (son) have a convergent subsequence? (c) Prove that if f is continuous everywhere on R, then it attains a maximum value, i.e. there exists a number wmax E R such that f('Tmax) Z f(33) for all .7: E R. (d) Write down an example of a continuous function on [R that is bounded above but does not attain a maximum value