Sequential Limit of a Sequence [Xn] Formed on Closed Subset of R

Limits of sequences behave well with respect to the usual arithmetic operations. If and , then , and, if neither b nor any is zero, . For any continuous function f, if then

• If for some constant c, then

. Proof: choose . We have that, for every , .

• If , then . Proof:

choose (the floor function). We have that, for every , .

• If when is even, and

when is odd, then . (The fact that whenever is odd is irrelevant.)

• Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence

converges to . Note that the decimal representation is the limit of the previous sequence, defined by.

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like , the number e. In these cases, one common approach is to find upper and lower bounds for the limit of the sequence (e.g., proving that ). Some other important properties of limits of real sequences include the following.

• The limit of a sequence is unique.

• provided

• If for all greater than some ,

then

• (Squeeze Theorem) If for all

, and , then .

• If a sequence is bounded and monotonic then it is convergent.

• A sequence is convergent if and only if every subsequence is convergent.

These properties are extensively used to prove limits without the need to directly use the cumbersome formal definition. Once proven that it becomes easy to show that , (), using the properties above.

The terminology and notation of convergence is also used to describe sequences whose terms become very large. A sequence is said to tend to infinity, written or if, for every K, there is an N such that, for every , ; that is, the sequence terms are eventually larger than any fixed K. Similarly, if, for

1. The limit of a convergent sequence is unique. 2. Every convergent sequence is bounded. This is a quite interesting result since it implies that if a sequence is not bounded, it is therefore divergent. For example, the sequence is not bounded, therefore it is divergent. 3. Any bounded increasing (or decreasing) sequence is convergent. Note that if the sequence is increasing (resp. decreasing), then the limit is the least-upper bound (resp. greatest-lower bound) of the numbers , for . 4. If the sequences and are convergent and and are two arbitrary real numbers, then the new sequence is convergent. Moreover, we have . It is also true that the sequence is convergent and . 5. If the sequence is convergent and and for any , then the sequence is convergent. Moreover, we have . The following examples will be useful to familiarize yourself with limit of sequences. Example: Show that for any number a such that 0 < a <1, we have .

obviously decreasing and bounded; hence it is convergent. Write . We need to show that L=0. We have , since the sequence is a tail of the sequence ; hence they have the same limit. But, using the previous properties, we get , which implies . Since , then we must have L=0. One may wonder, what happened to the sequence if a > 1? It is divergent since it is not bounded. This follows from and . Remark: Note that it is possible to talk about a sequence of numbers which converges to . Of course, we do reserve the word convergent to sequences which converges to a number; is not a number. The following shows the process: The sequence converges to (or, to ), if and only if, for any real number M > 0, there exists an integer , such that ). In particular, if as and for any , then we have A sequence which converges to is obviously not bounded. For example, we have for any a > 1. A sequence {xn}n∈N⊂X is said to converge towards x0∈X if for any ε>0 there is a natural number nε with the property that xn∈Bε(x0) for all n≥nε: Lim n→∞xn=x0⟺def∀ε>0∃ n ε ∈N;xn∈Bε(x0) for n≥nε. We then say that xn tends to x0 as n tends to infinity, written xn→x0(as n→∞), or xn→n→∞x0. The point x0 is called the limit of the sequence {xn}n∈N. Sequential limits are zero limits for the distance function .Since {dX(xn,x0)}n∈N is a sequence in R it is easily verified that xn→x0⟺dX(xn,x0)→0.

We say that f(x) converges to y0 in Y as x converges to x0 in X if for any ε>0 there exists δ>0 such that f(x)∈Bε(y0) when x∈Bδ(x0): Lim x→x0f(x)=y0⟺def∀ε>0∃δ>0;[dX(x,x0)<δ⟹dY(f(x),y0)<ε]. Equivalent ways of writing this are f(x)→y0 as x→x0, and f(x)→x→x0y0.

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at all points x0∈D, and simply continuous if its continuous on all of its domain. limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit is said to not exist. The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function. To say that means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p. The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts. Functions on the real line Suppose f : R → R is defined on the real line and p,L ∈ R. It is said the limit of f as x approaches p is L and written if the following property holds:

• For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f(x) − L | < ε.

Note that the value of the limit does not depend on the value of f(p), nor even that p be in the domain of f. A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real- for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ (a, b) implies | f(x) − L | < ε. Note that the limit does not depend on f(p) being well-defined. The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point. As discussed below this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.

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