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Unformatted text preview: hat N is a positive integer.
0 and an integer n
nx
.
1 n2x2
and we shall do so by giving an example of such a number . Now we want to prove that there exists a number N such that We define
To prove that there exists an integer n N
1 N2 1
2
1
2 2 . N such that
nx
1 n2x2 we make the observation that
Nx
1 N2x2 . 2. For every number x
0, 1 and every number 0 there exists a number 0 such that for every number
t
0, 1 satisfying t x  we have t 2 x 2  . Solution: We shall prove that this statement is true. Since we want to prove that something happens
for every number x
0, 1 , we begin as follows:
Suppose that x 0, 1 . Now we want to prove that something happens for every number
Suppose that 0 and so we continue: 0. We are now seeking a number 0 such that the inequality t 2 x 2  will hold whenever t
0, 1 we have
t x  . We observe first that whenever t
2t x .
t 2 x 2  t x t x  t x  t  x 
With this inequality in mind we define /2.
We see at once that whenever t 0, 1 and t
t2  x2  0, 1 and x  we have
2t x 2 2 . 3. For every number 0 and every number x
0, 1 there exists a number 0 such that for every number
t
0, 1 satisfying t x  we have t 2 x 2  . 24 Hint: The statement that appears here is identical to the one in Exercise 2.
4. For every number 0 there exists a number 0 such that for every number x
t
0, 1 satisfying t x  we have t 2 x 2  . 0, 1 and every number Solution: We shall prove that this statement is true. Since we want to prove that something happens
for every number
Suppose that 0 we begin: 0. We are now seeking a number 0 such that the intequality t 2 x 2  will hold for all numbers t and x
in the interval 0, 1 satisfying the condition t x  . To give an example of such a number we
continue:
Define /2.
We see at once that whenever t 0, 1 and x
t2 x2   0, 1 and t
x 2 2t x  we have
2 . 5. For every number 0 and every number x there exists a number 0 such that for every number t
satisfying t x  we have t 2 x 2  . Solution: The statement made in this exercise says a little more than the one in Exercise 2 but it is
still true. Since we want to prove that something happens for every number 0 we begin:
Suppose that 0. Now we want to prove that something happens for every number x and so we continue:
Suppose that x is a real number.
We are now seeking a number 0 such that for every number t satisfying t x  we have
t 2 x 2  . To help us find an example of such a number we observe first that if t x  1 x x 1 t x1 then, since
t  t x x t x  x  1 x  we have
t 2 x 2  t x t x  t x  t  x  t x  1 2x  With this inequality in mind we define to be the smaller of the two numbers 1 and / 1 2x  .
We observe that whenever a number t satisfies the inequality t x  we have
1 2x  .
t 2 x 2  t x  1 2x 
1 2x 
6. For...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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