How would you use induction to show that for $\displaystyle a_{i}\epsilon R$
$\displaystyle ||a_{1} + a_{2} + ... + a_{n}|| \leq ||a_{1}|| + ||a_{2}|| + ... + ||a_{n}|| $ ?
This seems like a very general induction proof. Thank you. Barton.
How would you use induction to show that for $\displaystyle a_{i}\epsilon R$
$\displaystyle ||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 $\displaystyle a_1, a_2 \in \mathbb{R} $:
$\displaystyle ||a_1+a_2||\le ||a_1||+||a_2||$
Now suppose that for some $\displaystyle k$ and for all $\displaystyle a_1, .. a_k \in \mathbb{R}$ that:
$\displaystyle ||a_1+ .. +a_k||\le ||a_1||+ .. + ||a_k||.$
Now consider:
$\displaystyle ||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:
$\displaystyle ||a_1+ .. +a_k+a_{k+1}||=||(a_1+ .. + a_k)+a_{k+1}|| \le ||a_1+ .. + a_k||+||a_{k+1}|| $ $\displaystyle \ \le ||a_1|| + .. + ||a_k||+||a_{k+1}||$
by the assumption.
You should now be able to put this all together to give the induction proof.
RonL