1.? ∈ 𝑆
2.? ∈ 𝑆with 𝑘 ≥ ? implies that n+1∈ 𝑆
4) Prove if x≥ 0and x≤ 𝜀for each𝜀 > 0, then x =0 Suppose 𝜀, ? ∈ ℝ, by O2, if 0≤ ? 𝑎?? x≤ 𝜀, then 0≤ 𝜀. But since 0 is strictly less than 𝜀
6) a. xy
≤
xy
(xy)
≤
xy
≤
xy
xy
≤
xy
xx
≤
yy if x
≥ 0

b. xy
< ?
xy
<
xy
<
c
Implies that xy
< ?
and x
<
c+ y
c. If xy
< 𝜀
and 0
< 𝜀
then xy=0
x=y
7) Prove that x
1
+x
2
+
…
.x
n

≤
x
1
+x
2
+
…
.x
n

??? ???? ? ∈
ℕ
Step 1:x
1

≤
x
1

Step 2: Assume that x
1
+x
2
+
…
.x
n

≤
x
1
+x
2
+
…
.x
n

??? ???? ? ∈
ℕ
Step 3: Then x
1
+x
2
+
…
.x
n
+x
n+1

≤
x
1
+x
2
+
…
.x
n
+x
n+1
.
Suppose x
1
+x
2
+
…
.x
n
 =a
Then a + x
n+1

≤
x
1
+x
2
+
…
.x
n
+x
n+1

By the triangle inequality, this is true
Extra Assigned Problems
You've reached the end of your free preview.
Want to read all 5 pages?
 Fall '08
 Staff
 7m