induction proof

**Barton** · August 4th 2008, 09:49 AM

How would you use induction to show that for $a_{i}\epsilon R$

$||a_{1} + a_{2} + ... + a_{n}|| \leq ||a_{1}|| + ||a_{2}|| + ... + ||a_{n}||$ ?

This seems like a very general induction proof. Thank you. Barton.

**CaptainBlack** · August 4th 2008, 11:16 AM

Originally Posted by Barton

How would you use induction to show that for $a_{i}\epsilon R$

$||a_{1} + a_{2} + ... + a_{n}|| \leq ||a_{1}|| + ||a_{2}|| + ... + ||a_{n}||$ ?

This seems like a very general induction proof. Thank you. Barton.

First poove as a base case that for all $a_1, a_2 \in \mathbb{R}$ :

$||a_1+a_2||\le ||a_1||+||a_2||$

Now suppose that for some $k$ and for all $a_1, .. a_k \in \mathbb{R}$ that:

$||a_1+ .. +a_k||\le ||a_1||+ .. + ||a_k||.$

Now consider:

$||a_1+ .. +a_k+a_{k+1}||=||(a_1+ .. + a_k)+a_{k+1}|| \le ||a_1+ .. + a_k||+||a_{k+1}||$

by the base case, and so:

$||a_1+ .. +a_k+a_{k+1}||=||(a_1+ .. + a_k)+a_{k+1}|| \le ||a_1+ .. + a_k||+||a_{k+1}||$ $\ \le ||a_1|| + .. + ||a_k||+||a_{k+1}||$

by the assumption.

You should now be able to put this all together to give the induction proof.

RonL

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