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# Definition of Limit

Visual explanation by Daisy of the convergence of a sequence to a limit.

Below we outline a high-level schematic for proofs in these notes. More specific outlines follow, and then several examples are provided in subsequent pages. As with most proofs, a rough outline is to take the following approach.

- 1.State relevant definitions and the math statement we must verify (
*e.g.*an inequality). - 2.Find relationships (e.g. equalities) or theorems that can tie things we know (from the assumptions) to what we want.
- 3.Combine results in Step 2 to verify the statement of Step 1.

A sequence

$\mathsf{\{s_n\}}$

of real numbers converges to a limit $\mathsf{s}\in\mathbb{R}$

provided, for all $\mathsf{\epsilon>0}$

, there is a natural number $\mathsf{N\in\mathbb{N}}$

such that $\mathsf{|s_n-s|\leq \epsilon}$

, for all $\mathsf{n\geq N.}$

Suppose we must prove

$\{a_n\}$

converges to a limit $L$

. Then we can take the following steps.**Step 1.**Write an introduction of what must be shown like the example below.

Let

$\varepsilon > 0$

be given. It suffices to show there exists an index $N\in\mathbb{N}$

such that $|a_n - L|\leq\varepsilon$

for all $n\geq N.$

**Step 2.**Do scratch work to find an upper bound for

$|a_n-L|$

in terms of something we know how to bound by $\varepsilon$

(*e.g.*

$1/n$

as in the Archimedean examples below). Then choose a large $N$

that makes the upper bound on $|a_n - L| \leq \varepsilon$

for $n\geq N$

.**Step 3.**Tie everything together by combining the results to verify

$|a_n-L|\leq\varepsilon$

for all $n\geq N.$

Slides_Seq_Limit_Def.pdf

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