math104-s21:s:johndufek
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math104-s21:s:johndufek [2021/05/07 01:44] 73.71.56.149 |
math104-s21:s:johndufek [2026/02/21 14:41] (current) |
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| + | **Limit Proofs of Sequences** | ||
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| + | . A very helpful technique for proving the limit of some sequence where it is really hard to isolate n can be the following: \\ | ||
| + | //Example: say you want to prove the limit of $\frac{4n^3 + 3n}{n^3 - 6} => \frac{3n+24}{n^3 - 6} = 4 $ . It would be hard to isolate $n$ to find such an $N > 0$. \\ | ||
| + | Hence bound the sequence by something easier i.e. $3n + 24 \leq 27n$. Thus $\frac{27n}{n^3 - 6} < \epsilon$. Building off this technique we know the denominator $n^3 - 6 \geq \frac{n^3}{2}$ . Hence $\frac{27n}{n^3 / 2} < \epsilon => \frac{54}{n^2} < \epsilon$ .Then from here you can easily find an $N > 0$ s.t. the limit definition holds. This technique is displayed in Ross page 40 - 41.// \\ | ||
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| + | //See Ross textbook chapter 9 for useful limit properties and theorems.// | ||
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| + | **Sequences**\\ | ||
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| $(b)\ d(p,q)\ =\ d(q,p)$ \\ | $(b)\ d(p,q)\ =\ d(q,p)$ \\ | ||
| $( c )\ d(p,q)\ \le \ d(p,r)\ +\ d(r,q),\ for\ any\ r\ \in \ X$ \\ | $( c )\ d(p,q)\ \le \ d(p,r)\ +\ d(r,q),\ for\ any\ r\ \in \ X$ \\ | ||
| - | $d(x,y)\ =\ |x-y|\ (x,y\ \in \ R^k \ )$ \\ | + | $d(x,y)\ =\ |x-y|\ (x,y\ \in \ \R^k \ )$ \\ |
| . A K-cell is convex and compact\\ | . A K-cell is convex and compact\\ | ||
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| . **Thm 2.35 corr. ~ Rudin** If $f$ is closed and $K$ is compact, then $F \cap K$ is compact.\\ | . **Thm 2.35 corr. ~ Rudin** If $f$ is closed and $K$ is compact, then $F \cap K$ is compact.\\ | ||
| . **Thm 2.37 ~ Rudin** If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$. \\ | . **Thm 2.37 ~ Rudin** If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$. \\ | ||
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| . **Thm 2.42 ~ Rudin** Every bounded, infinite set in $\R ^k$ has a limit point in $\R ^k$. \\ | . **Thm 2.42 ~ Rudin** Every bounded, infinite set in $\R ^k$ has a limit point in $\R ^k$. \\ | ||
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| . **Thm 4.14 ~ Rudin** if $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$, then $f(x)$ is compact. \\ | . **Thm 4.14 ~ Rudin** if $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$, then $f(x)$ is compact. \\ | ||
| . **Thm 4.17 ~ Rudin** if $f$ is a continuous one to one mapping of a compact space $X$ ONTO a metric space $Y$, then the inverse mapping $f^{-1}$ is a continuous mapping of $Y$ onto $X$. \\ | . **Thm 4.17 ~ Rudin** if $f$ is a continuous one to one mapping of a compact space $X$ ONTO a metric space $Y$, then the inverse mapping $f^{-1}$ is a continuous mapping of $Y$ onto $X$. \\ | ||
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| . **Thm 4.19 ~ Rudin** If $f$ is a continuous mapping of compact metric space $X$ INTO a metric space $Y$, then $f$ is uniformly continuous on $X$. \\ | . **Thm 4.19 ~ Rudin** If $f$ is a continuous mapping of compact metric space $X$ INTO a metric space $Y$, then $f$ is uniformly continuous on $X$. \\ | ||
| . **Thm 4.22 ~ Rudin** If $f$ is a continuous mapping of a metric space $X$ INTO a metric space $Y$, an dif $E$ is a connected subset of $X$, then $f(E)$ is connected.\\ | . **Thm 4.22 ~ Rudin** If $f$ is a continuous mapping of a metric space $X$ INTO a metric space $Y$, an dif $E$ is a connected subset of $X$, then $f(E)$ is connected.\\ | ||
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| 2) | 2) | ||
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| + | **Limits**\\ | ||
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| + | 1) How can we know/ differentiate certain approaches to particular limits of sequences that satisfy L' | ||
math104-s21/s/johndufek.1620351841.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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