Prove that if a n= 1/ n, then ( a n) → 0. for every positive real ε, there exists N, such that whenever n>N, we have |a n – L| N, we have | a n– K| M, we have | a n– L| max( M, N) such thatĮxample 1.We say that (a n) converges to a real number L (written a n → L) if: One can think of it as a function N → R, where N is the set of positive integers. Our focus here is to provide a rigourous foundation for the statement “sequence ( a n) → L as n → ∞”.ĭefinition (Convergence). ![]() Definition of ConvergenceĪ sequence in R is given by ( a 1, a 2, a 3, …), where each a i is in R. Thus, L = sup(S) iff (i) L is an upper bound, (ii) for each ε>0, there exists x in S, x > L-ε. Since L-ε is not an upper bound, there exists x in S, x > L-ε. the set of real (or rational) numbers 0 0).
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