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\noindent {\bf Problem statement} a) Prove: if $0\le x \le 10$, then
$0\le \sqrt{x+1} \le 10$.

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\noindent {b) Prove: if $0\le u \le v \le 10$, then $0 \le \sqrt{u+1} \le
\sqrt{v+1} \le 10$.

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\font\ssb=cmssbx10.tfm
\noindent {c) Consider the following recursively defined sequence:

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\qquad $a_1= 0.3$; $a_{n+1} = \sqrt{a_n + 1}$ for $n>1$.

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\noindent and compute its first five terms. Prove that this sequence
converges. You should mention a specific {\ssb THEOREM} from section
10.1 which, together with the algebraic results of parts a) and b),
will guarantee convergence. 

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\noindent {d)} What is the exact limit of the sequence defined in c)?
(Square the recursive equation and take limits using some limit
theorems.)

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\noindent {\bf Note} Many sequences encountered in ``real life'' are
recursively defined, and the method in d) is often used. On many
calculators, you can define the function $\sqrt{x+1}$ and then use the
{\tt ANS} key repeatedly to calculate many terms of the
sequence easily.











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