Finding the limit of sequence by the pinch theorem Using pinch theorem to find the limit of sequence (n!) / (n power of n) when n tends to positive infinity

Function limit and sequence limit (Heine's theorem)
On the sufficiency of its proof, I can't understand the proof of Heine's theorem in Baidu Encyclopedia
Lim [x - > A] f (x) = b = = > Lim [n - > ∞] f (an) = B is defined by function limit: if e > 0, there exists d > 0, if | x-a | a] f (x) is not B, then there exists e > 0. For any d > 0, there exists a certain X: satisfy | x-a | e, and then the contradiction can be deduced by using the definition of sequence limit of LIM [n - > ∞] f (an) = B
Among them, "using the limit definition of LIM [n - > ∞] f (an) = B to deduce the contradiction." I don't know who can write it in detail here. Please explain it to me. Thank you

Key: any sequence an proof: find a sequence limit definition that does not satisfy Lim [n - > ∞] f (an) = B. prove: if Lim [x - > A] f (x) is not B, then there exists E > 0, for any D1 > 0, there exists a x1, and X1 is not equal to a: satisfy | x1-a | e, note A1 = x1, there also exists E > 0, for any D2 > 0, we might as well take D2 = D1 /

Solving the limit problem of sequence
The limits of sequence xn and sequence yn are a, B respectively, and a is not equal to B. what are the limits of sequence x1, Y1, X2, Y2, X3, Y3?
It's a concrete process

Proof: we take the subsequences xn and yn of the sequence x1, Y1, X2, Y2, X3, Y3
Because limxn = a, limyn = B, and a is not equal to B
So the sequence x1, Y1, X2, Y2, X3, Y3. Does not converge, that is divergence. Then the limit does not exist
(Note: because any subsequence of a convergent sequence converges and converges to the same limit)

Inequality preserving property of sequence limit!
In mathematical analysis, when proving, does the inequality preserving property of sequence limit always have an equal sign, and some do not have an equal sign? It is said that in precise consideration, the equality can be excluded, please give examples

Let limxn = x, limyn = y, if x > y, then there is n, for any n, when n > N, there is xn > YN, for example: xn = 1-1 / N, yn = 1 / N, limxn = 1, limyn = 0,1 > 0, remove n = 2, then when n > N, there is xn > YN, let limxn = x, limyn = y, if for every n, there is xn > YN, there is limxn > = limyn, then the equal sign cannot be removed, for example: X

Is a bounded sequence a sequence with limits? Why

No. boundedness and limit are two concepts. A bounded sequence is an exponential sequence, each term of which does not exceed a fixed interval. It is divided into upper bound and lower bound. Suppose there is a fixed value a and any n has an = B, the sequence an is said to have a lower bound B. If there is a and B at the same time, the value of the sequence an is in the interval [a, b], the sequence an is bounded and bounded

If a sequence has a limit, it must be bounded. Why?

If a sequence has a limit, it must be bounded
prove:
If an → a,
Then there is a natural number n for all E > 0,
When n > N, then | an-a | n, then a-e

How to prove that a sequence has a limit, then it must be bounded
It is said in a book that because all the terms of a ruler fall in the neighborhood of a, there are only a limited number of points outside the neighborhood, so it is proved. But does a limited number of points necessarily mean that they occupy a limited space, and can't the points be infinite
2. Prove that two parallel lines in space are on the same plane in elementary geometry

There is something wrong with your understanding. A finite number of points are definite points, and the distance is always limited. Therefore, two lines parallel to the X axis can always be found to clamp them, so they must be bounded. The ordinates corresponding to the two lines are the upper and lower bounds

Group the items of the equal ratio sequence {an = 3 ^ n-1} into {1}, {3,9}, {27,81243} according to the following rules What is the sum of each number in group 6

The nth group of the sequence consists of N numbers
Therefore, the sixth group is composed of six numbers, and the front five groups have a total of 1 + 2 + 3 + 4 + 5 = 15 numbers, so the first number of the sixth group is 3 ^ 15
The sum of each number in group 6 is 3 ^ 15 + 3 ^ 16 +... + 3 ^ 20
=(3^21-3^15)/2=5223002148
(this question should be in exponential form, otherwise it will be very tangled.)

Which of the following numbers are sequence - 1,3, - 9,27 , items? A, 81 B, - 81 C, 243 D, - 243

BC
-1,3,-9,27,-81,243

1, - 1 / 3,1 / 9, - 1 / 27,1 / 81 ·································································,

An=(-1)^(n-1)*(1/3)^(n-1)

Finding the limit of sequence by the pinch theorem Using pinch theorem to find the limit of sequence (n!) / (n power of n) when n tends to positive infinity