Page 206 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Now, consider

n n+1

and x m , n >N.
x m
m=1 m=1

They both satisfy

n n+1

x m − L <, x m − L <.

m=1 m=1

This means that they are both at a distance less than from L . What’s the difference

n+1 n

x m − x m =?
m=1 m=1

Try going back to the definition of convergence for the series, and because > 0, then

> 0
2

also. Now you can put 0 in the disguise −L + L and

n n+1 n n+1

x m .

x m − x m − L + L −
x m =
m=1 m=1 m=1 m=1

Next you can use the triangle inequality.
√
• Hint for # 5: Remember the 2 problem? I hope it’s not becoming a mathematical
monster under your mathematical bed! Try a proof by contradiction using the FTA and the
definition of p root: the p root of q is the unique real number x such that
th
th
p
x = q.

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