A sequence of bounded functions fn​:S→R converges uniformly to f:S→R, if and only if limn→[infinity]​∥fn​−f∥u​=0, where ∥f∥u​:=sup{∣f(x)∣:x∈S}. (5.2) Consider the sequence (fn​) defined by fn​(x)=1+nxnx​, for x≥ 0. (5.2.1) Find f(x)=limn→[infinity]​fn​(x). (5.2.2) Show that for a>0,(fn​) converges uniformly to f on [a,[infinity]). (5.2.3) Show that (fn​) does not converge uniformly to f on [0,[infinity]). (5.3) Suppose that the sequence (fn​) converges uniformly to f on the set D and that for each n∈N,fn​ is bounded on D. Prove that f is bounded on D. (5.4) Give an example to illustrate that the pointwise limit of continuous functions is not necessarily continuous.