tags:
  - RealAnalaysis

Note: I am using the Forrest Textbook for this. The order and subsequently numbering of the Theorems has no meaning. These are the relatively less important Theorems but are needed in proving the more important ones.

Theorem 1
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Forrest_M147CN_F20 (1).pdf > page=114

Assume that converges. Then is bounded when viewed as a subset of .

PROOF

We know that there exists a so that if , then

The Triangle Inequality then shows that

for all . Let

then

for all .

Theorem 2
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**Let be a sequence with Assume that . Then

Very very similar proof (> not ≥) can be found in the HW assignment 1 for 147.
Crowdmark.pdf > page=8

is convergent, .

Let .

For contradiction, let's assume .

Since , .

Thus, .

Setting ():

This is a contradiction since and .

Therefore, our assumption that is wrong, and .

Theorem 3
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Forrest_M147CN_F20 (1).pdf > page=117

Assume that and are two sequences and that Assume also that exists. Then,

Proof

Using Limits of a Sequence > Property 4

Since
,

We can similarly use the same technique used for Limits of a Function > Arithmetic Rules for Limits of Functions to prove this for limits of a function.

Theorem 1...

Theorem 2...

Theorem 3...

Theorem 1
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Theorem 2
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Theorem 3
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