Assume
![\dfrac1{a_n}](https://tex.z-dn.net/?f=%5Cdfrac1%7Ba_n%7D)
is not bounded, i.e. there are no
![u,\ell](https://tex.z-dn.net/?f=u%2C%5Cell)
for which
![\ell\le\dfrac1{a_n}\le u](https://tex.z-dn.net/?f=%5Cell%5Cle%5Cdfrac1%7Ba_n%7D%5Cle%20u)
for all
![n](https://tex.z-dn.net/?f=n)
.
Now,
![a_n\to k\neq0](https://tex.z-dn.net/?f=a_n%5Cto%20k%5Cneq0)
is to say that for any
![\varepsilon>0](https://tex.z-dn.net/?f=%5Cvarepsilon%3E0)
, we can find a large enough
![N](https://tex.z-dn.net/?f=N)
such that
![|a_n-k|](https://tex.z-dn.net/?f=%7Ca_n-k%7C%3C%5Cvarepsilon)
whenever
![n\ge N](https://tex.z-dn.net/?f=n%5Cge%20N)
. Simultaneously, this means that
![a_n](https://tex.z-dn.net/?f=a_n)
is bounded.
Let's suppose without loss of generality that
![a_n\neq0](https://tex.z-dn.net/?f=a_n%5Cneq0)
for any
![n](https://tex.z-dn.net/?f=n)
. (Note that if
![a_n=0](https://tex.z-dn.net/?f=a_n%3D0)
for some finite number of values of
![n](https://tex.z-dn.net/?f=n)
, we can simply remove them from consideration.)
So we have
![|a_n-k|=\left|a_n\left(1-\dfrac k{a_n}\right)\right|=|a_n|\left|1-\dfrac k{a_n}\right|](https://tex.z-dn.net/?f=%7Ca_n-k%7C%3D%5Cleft%7Ca_n%5Cleft%281-%5Cdfrac%20k%7Ba_n%7D%5Cright%29%5Cright%7C%3D%7Ca_n%7C%5Cleft%7C1-%5Cdfrac%20k%7Ba_n%7D%5Cright%7C%3C%5Cvarepsilon)
Because
![a_n](https://tex.z-dn.net/?f=a_n)
is bounded, we know there is some
![\alpha](https://tex.z-dn.net/?f=%5Calpha)
such that
![|a_n|\le\alpha](https://tex.z-dn.net/?f=%7Ca_n%7C%5Cle%5Calpha)
for all
![n](https://tex.z-dn.net/?f=n)
. Now,
![|a_n|\left|1-\dfrac k{a_n}\right|\le\alpha\left|1-\dfrac k{a_n}\right|](https://tex.z-dn.net/?f=%7Ca_n%7C%5Cleft%7C1-%5Cdfrac%20k%7Ba_n%7D%5Cright%7C%5Cle%5Calpha%5Cleft%7C1-%5Cdfrac%20k%7Ba_n%7D%5Cright%7C%3C%5Cvarepsilon)
![\left|1-\dfrac k{a_n}\right|](https://tex.z-dn.net/?f=%5Cleft%7C1-%5Cdfrac%20k%7Ba_n%7D%5Cright%7C%3C%5Cdfrac%5Cvarepsilon%5Calpha)
![-\dfrac\varepsilon\alpha](https://tex.z-dn.net/?f=-%5Cdfrac%5Cvarepsilon%5Calpha%3C1-%5Cdfrac%20k%7Ba_n%7D%3C%5Cdfrac%5Cvarepsilon%5Calpha)
![-1-\dfrac\varepsilon\alpha](https://tex.z-dn.net/?f=-1-%5Cdfrac%5Cvarepsilon%5Calpha%3C-%5Cdfrac%20k%7Ba_n%7D%3C-1%2B%5Cdfrac%5Cvarepsilon%5Calpha)
![1-\dfrac\varepsilon\alpha](https://tex.z-dn.net/?f=1-%5Cdfrac%5Cvarepsilon%5Calpha%3C%5Cdfrac%20k%7Ba_n%7D%3C1%2B%5Cdfrac%5Cvarepsilon%5Calpha)
![\dfrac1k-\dfrac\varepsilon{\alpha k}](https://tex.z-dn.net/?f=%5Cdfrac1k-%5Cdfrac%5Cvarepsilon%7B%5Calpha%20k%7D%3C%5Cdfrac%201%7Ba_n%7D%3C%5Cdfrac1k%2B%5Cdfrac%5Cvarepsilon%7B%5Calpha%20k%7D)
But we initially assumed that
![\dfrac1{a_n}](https://tex.z-dn.net/?f=%5Cdfrac1%7Ba_n%7D)
is unbounded, so the above is impossible. Thus
![\dfrac1{a_n}](https://tex.z-dn.net/?f=%5Cdfrac1%7Ba_n%7D)
must be bounded.
Answer:
A) Bobby's overall grade is 70.4%
Step-by-step explanation:
To solve this problem we must calculate the percentages each grade represents from the overall grade. Therefore:
assignment = 85*30% = 85*30/100 = 25.5%
tests = 72*20% = 72*20/100 = 14.4%
exam = 61*50% = 61*50/100 = 30.5%
total = assignment + tests + exam
total = 25.5 + 14.4 + 30.5 = 70.4
Bobby's overall grade is 70.4%. Therefore the correct answer is A).
Answer:
Here's one way to do it
Step-by-step explanation:
1. Solve the inequality for y
5x - y > -3
-y > -5x - 3
y < 5x + 3
2. Plot a few points for the "y =" line
I chose
![\begin{array}{rr}\mathbf{x} & \mathbf{y} \\-2 & -7 \\-1 & -2 \\0 & 3 \\1 & 8 \\2 & 13 \\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brr%7D%5Cmathbf%7Bx%7D%20%26%20%5Cmathbf%7By%7D%20%5C%5C-2%20%26%20-7%20%5C%5C-1%20%26%20-2%20%5C%5C0%20%26%203%20%5C%5C1%20%26%208%20%5C%5C2%20%26%2013%20%5C%5C%5Cend%7Barray%7D)
You should get a graph like Fig 1.
3. Draw a straight line through the points
Make it a dashed line because the inequality is "<", to show that points on the line do not satisfy the inequality.
See Fig. 2.
4. Test a point to see if it satisfies the inequality
I like to use the origin,(0,0), for easy calculating.
y < 5x + 3
0 < 0 + 3
0 < 3. TRUE.
The condition is TRUE.
Shade the side of the line that contains the point (the bottom side).
And you're done (See Fig. 3).
The answer is b because its hard to explain