1 Let (an) nen be a sequence. (a) Prove that if lim an = L exists, then every subsequence of (an)ner tends to L. 1-+0o (b) Prove that if every subsequence of (an )nel is convergent, then (an)EN is convergent. 2 (a) Prove that if (bn)ner converges to 0 and by 2 0 for each n E N, then (Vb,) nel tends to 0. (b) Prove that if (an)nel converges to A > 0 and an 2 0 for each n E IN, then (VA+ Van )nen is bounded. 3 Find the limits of the following sequences: (a) (VAn' +17 ) neN' (b) (n - 2n2-An+1) 1+1 ) MEN ' EN ' ( C ) ( Vn 2 + 5n - n ) new 4 Explain why the following inequalities are all valid: 1 S n/ 0 there exist natural numbers k, l, m such that |2 - (2 - 2mm)