
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</= 2a+1
iii) a[n]^2  a^2 </= a[n] a(2a+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

This is a basic fact: $\displaystyle \left a \right  \left b \right \leqslant \left {\left a \right  \left b \right} \right \leqslant \left {a  b} \right$.
From that you can prove many things.