"Finite Changes to an Infinite Sequence Do Not Affect its Limit"
What is the limit of the sequence:
{1, 1, 1, .... 1, 2, 2, 2, ....}
(first billion terms are "1" and the rest are "2")

The limit of this seuqence equals _____ Why?____________________________________

What is the limit of the sequence whose first trillion terms are the corresponding digits of pi, and the rest of whose terms are 3.14, and why? ______________________________.

Suppose the limit of {a_n} is L, and the limit of {b_n} is M.

Now let {c_n} be such that c_n=a_n for n=1,2,...1000 and such that c_n= b_n for n=1001,1002,... What is the limit of the sequence {c_n}? Why?

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What if c_n=a_n for n even, and c_n=b_n for n odd?

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What Theorem do these examples suggest? State your own theorem in your own (careful, mathematically precise) terms:

[Your name]:     _______________'s Theorem:

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The Sandwich Theorem: If the sequences {a_n} and {c_n} both converge, both approach (converge to) the same limit L, and if for every n one has

then (a) it's guaranteed that {b_n} converges and (b) it will converge to L.

Let b_n= sin(n)/n. Prove {b_n} converges, and what it converges to, using the Sandwich Theorem.

Give a graph of your three functions (sequences):

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How does this support your calculation? And how does it support the Sandwich Theorem more generally?

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Theorem on Monotone and Bounded Sequences

Use your books if necessary.

A sequence {a_n} is increasing if
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while it is said to be decreasing if
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A sequence {a_n} is monotone if
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A sequence {a_n} is bounded above if
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while a sequence {a_n} is bounded below if
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A sequence {a_n} is bounded if
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Theorem:

If a sequence {a_n} is __________ and __________ then ____________________.

Application:

Prove that the sequence {0.9, 0.99, 0.999, 0.9999, ....} must converge (to some number).

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What do you think it converges to? __________.

(Optional -- for students who have had delta-epsilon definitions:)

How would you "prove" it using the technical definition of limits of sequences?
(ask your instructor for the N-epsilon definition if you need to).

Your ideas: