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Math Help - A proof with induction, really needed!

  1. #1
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    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
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  2. #2
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    I will assume that the given means: \left| {a_1 } \right| \le 1\,\& \,\forall n\left[ {\left| {a_n  - a_{n + 1} } \right| \le 1} \right].

    Note that: \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 \left| {a_{n - 1} } \right| \le n - 1 it follows that
    \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.
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