A proof with induction, really needed!

• Mar 5th 2008, 05:53 AM
ml_lulu
A proof with induction, really needed!
If |a1|<=1 and |an - an-1|<=1 , then |an|<=n

I started with |ak - ak-1|<=1 and used the triangle inequality but I get two things that are grater than or equal to |an - an-1| ! If I assume that |ak| + |ak-1|<=1 and then take for granted that if thats true then |ak+1| + |ak|<=1 then I can proove the whole thing, but I donīt really think Iīm doing it right. Please help me! Its going to appear in a test... Thanks :)
• Mar 5th 2008, 09:34 AM
Plato
I will assume that the given means: $\displaystyle \left| {a_1 } \right| \le 1\,\& \,\forall n\left[ {\left| {a_n - a_{n + 1} } \right| \le 1} \right]$.

Note that: $\displaystyle \left| {a_2 } \right| - \left| {a_1 } \right| \le \left| {a_2 - a_1 } \right| \le 1 \Rightarrow \quad \left| {a_2 } \right| \le 1 + \left| {a_1 } \right| \le 2$.

If we know that $\displaystyle \left| {a_{n - 1} } \right| \le n - 1$ it follows that
$\displaystyle \left| {a_n } \right| - \left| {a_{n - 1} } \right| \le \left| {a_n - a_{n - 1} } \right| \le 1 \Rightarrow \quad \left| {a_n } \right| \le 1 + \left| {a_{n - 1} } \right| \le 1 + (n - 1) = n$.