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When I initially moved from Applied Mathematics background to Pure Mathematics (Graduate school), things were very tough in terms of proofs but they have gotten much better now. At those times, I couldn't explain what the following meant in proofs;

1. Let $epsilon>0$ be given;


2. Let $ninBbb{N}$ be fixed;


3. Let $xinBbb{R}$ be fixed but arbitrary;


4. Take $n=N+1,$ for some $NinBbb{N}$;


5. In particular, for $epsilon=1/2$;


6. Let $epsilon'>0$, for $epsilon=epsilon'/3$
   


7. Such that;


8. Whenever,


9. For each;


10. For all, and many more.

Could you please, explain these in details, perhaps, with some examples? That is, when to use them and how to... If you have other examples, I'd be glad to learn from you. Thanks

1. Let $epsilon>0$ be given; Take for example, Prove that $1/nto 0,$ as $ntoinfty.$

PROOF

Let $epsilon>0$ be given (it means give me a specific positive nuimber, say $epsilon$, then I'd prove to you that$ ;forall; epsilon>0$, $1/n<epsilon$ for large n). Hence, given any $epsilon>0$, choose $N=left[1/epsilon+1right]$, then,
begin{align} dfrac{1}{n}leq dfrac{1}{N}<epsilon,;forall; ngeq N. end{align}
This implies that for all $epsilon>0$,
begin{align} dfrac{1}{n}<epsilon,;forall; ngeq N. end{align}

2. Let $ninBbb{N}$ be fixed;

Pick a particular $ninBbb{N}$ for the whole proof which cannot be changed in the course of proving.

EXAMPLE 2

Prove that for bounded positive real sequences ${a_n}_{ninBbb{N}}$
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}

PROOF

Let $epsilon>0$ be given and $beta:=limsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| $, then there exists $N$ such that
begin{align} left|dfrac{a_{n+1}}{a_n} right|<beta+epsilon,;;forall;ngeq N. end{align}
Let $ngeq N$ be fixed,
begin{align} left|dfrac{a_{n}}{a_N} right|=left|dfrac{a_{n}}{a_{n-1}} right|cdot left|dfrac{a_{n-1}}{a_{n-2}} right|cdotsleft|dfrac{a_{N+1}}{a_{N}} right|<left(beta+epsilonright)^{n-N}end{align}
This implies
begin{align} sqrt[n]{left|a_{n}right|}<sqrt[n]{left|a_N right|}left(beta+epsilonright)^{1-N/n}end{align}
Taking $limsup$,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqbeta+epsilonend{align}
Since $epsilon>0$ was arbitrary,
begin{align} limsup_{ntoinfty} sqrt[n]{|a_n|}leqlimsup_{ntoinfty} left|dfrac{a_{n+1}}{a_n} right| end{align}

3. Such that, arbitrary

EXAMPLE 3

Prove that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous if

begin{align} forall;lambdain Bbb{R},;;f^{-1}left((lambda,infty] right);;text{is open in};X. end{align}
PROOF

Let $lambdain Bbb{R}$ and $x_0in X$ be arbitrary, such that $lambda<f(x_0)$. Then, begin{align} x_0in f^{-1}left((lambda,infty] right). end{align} Take begin{align} V=f^{-1}left((lambda,infty] right), ;;text{where };;Vin Uleft(x_0 right).end{align} Let $xin V$, then $f(x)in (lambda,infty] ,$ which implies that $f(x)>lambda.$ Hence, for all $lambdain Bbb{R}$ and $x_0in X$ such that $lambda<f(x_0)$, $f(x)>lambda,;forall;xin V.$ This implies that that $f:Xtooverline{Bbb{R}}$ is lower semi-continuous.

4. Fixed but arbitrary, fixed, such that and whenever.

EXAMPLE 4

Prove that $|x|$ is continuous on $Bbb{R}$.

Let $epsilon>0$ be given, $xin Bbb{R}$ be fixed but arbitrary and $x_0in Bbb{R}$ be fixed, such that $|x-x_0|<delta,$ then
begin{align} left| |x|-|x_0| right| leq left| x-x_0 right|<delta.end{align}
So, given any $epsilon>0$, choose $delta=epsilon,$ then
begin{align} left| |x|-|x_0| right|<epsilon,;;textbf{whenever};;left| x-x_0 right|<deltaend{align}
I pick any $xinBbb{R}$ for the proof but it will be fixed throughout the proof.