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Long Him Lui Sequences are a typical area of study in upper secondary mathematics. Typically, when given a sequence, we are interested in learning about the limit of a sequence. Take the following sequence (denoted x_n), as well as the following claim: with N denoting the set of natural numbers, and the symbol ϵ meaning an element of. This can be done intuitively, which is the approach in secondary school. As n increases, the value of the reciprocal of n decreases. Since both 1 and n are positive numbers, the limit will not tend to a negative number. Therefore, the limit will lie between 0<lim x_n<1. Imagine splitting a whole pizza into more and more slices. The size of each slice of pizza will gradually decrease as the number of slices increases, so when there are infinitely many slices, the size of the slice will be extremely small; nonzero but extremely close to 0. Today, I will be discussing the rigorous definition of the limit of a sequence, and showing how it can be applied to determine the limit of a sequence. The definition is as follows: a sequence <x_n> converges to a number l (or a limit l), where l ϵ R, (R denotes the set of real numbers) if the following property holds: The V-looking symbol denotes for all, and the backwards E denotes there exists. In words, this means that the sequence <x_n> converges to a limit l that is an element of the set of real numbers if given any ϵ > 0, so we can find a number N which is an element of the set of real numbers such that all terms of the sequence with n>N are inside a “neighborhood” (which we denote ϵ) of the value of the limit l. This definition is easier understood with the help of a diagram: Removing the modulus, the statement |x_n - l| states that the value of |x_n - l| lies between positive and negative ϵ. Hence: Therefore, the definition says that x_n converges to l for any specified ϵ that is small, and there is a term x_N of the sequence such that all subsequent terms lie in between l and ϵ. Looking at the diagram above, that value of N that would satisfy the given ϵ would be N=7, since all subsequent terms n>7 of the sequence all lie within the neighborhood of ϵ. For smaller neighborhoods, the value of N would be bigger so that all subsequent terms will be within that smaller neighborhood. This shows that as the distance from the limit to positive or negative ϵ decreases, there will always exist a point N such that all the subsequent points lie in the ϵ neighborhood. This implies that the sequence is tending towards that point, but will never reach it, because the value of can get infinitely small. So using our definition of the convergence of a sequence to limit l, we can solve the limit of: We know that the sequence tends to 0, so we can substitute l=0 and x_n=1/n: By the definition, there exists an N that is an element of Z, (Z denoting the set of integers), with N>1/ϵ: (--> denotes “implies that”) And we have returned to our original claim with the definition.
Thus, for every ϵ>0, a suitable N can be found. This specific case says that N is any integer exceeding 1/ϵ, and this concludes the proof. This definition can be applied to any sequence as long as it converges to a single specific value. If the sequence converges to positive or negative infinity, we say that the sequence diverges.
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