write handwritten proof

3. We will denote the set of positive integers by Zt. Let {n)n=1 1 be a sequence of real numbers. The alternating decomposition of {In }n=1 ", consists of the sequences { An}, and {Bn n=1 where ux = ug'I-ux = "y '+Z ) uA Prove the following lemma. Lemma: If . the sequences { An} In=1 and { Bnin ) are an alternating decomposition of a sequence { n )n=1 . { An)n= and { Bn)n= are both convergent to the same limit THEN the sequence {On} is also convergent to this limit. Hint: Use the definition of limit. Note that the sequence {n} is a general sequence of real number, unlike the other questions where sequences are positive