# Sequences and proofs

• May 20th 2009, 04:13 PM
mitch_nufc
Sequences and proofs
I have a sequence {a[n]} from n=1 to infinity, a is a real number

Show that if |a[n] -a|<1

then

i) |a[n]|</= |a| + 1
ii) |a[n] + a|</= 2|a|+1
iii) |a[n]^2 - a^2| </= |a[n] -a|(2|a|+1)

How are these done? I'm familiar with the triangle inequality and such, but these don't seem familiar. Perhaps I'm missing something basic or elementary? Any comments welcome, cheers
• May 20th 2009, 04:28 PM
Plato
This is a basic fact: $\left| a \right| - \left| b \right| \leqslant \left| {\left| a \right| - \left| b \right|} \right| \leqslant \left| {a - b} \right|$.

From that you can prove many things.