Math CentralQuandaries & Queries


Question from James, a teacher:

I need to prove a problem by induction regarding the Triangle Inequality. The problem is

abs(a1 + a2 +...+an) <= abs(a1) + abs(a2) +...+ abs(an).


For problems like this, the key idea is to write

abs(a1 + a2 + ... + an) = abs(a1 + a2 + ... + (an-1 + an)).

The expression on the right hand side is a sum of n-1 terms. That lets you use the induction hypothesis to get

<= abs(a1) + abs(a2) +...+ abs(an-1+an)

and then apply the base case to the last term.

Good luck.

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