Question

Use the Triangle Inequality and Mathematical Induction to show that\[\left|\sum_{k=1}^{n}a_k\right|\leq\sum_{k=1}^{n}|a_k|.\]Well, from the Triangle Inequality, we know that\[\left|\sum_{k=1}^{n}a_k\right|=|a_1+a_2+...+a_n|\leq|a_1|+|a_2|+...+|a_n|=\sum_{k=1}^{n}|a_k|,\]and by Mathematical Induction, we see that\[\left|\sum_{i=1}^{1}a_i\right|=|a_1|=\sum_{i=1}^{1}|a_i|,\]assume it's true for \(k\), and see that\[\left|\sum_{i=1}^{k+1}a_i\right|=|a_1+a_2+...+a_{k+1}|\leq|a_1|+|a_2|+...+|a_{k+1}|=\sum_{i=1}^{k+1}|a_i|,\]as required.

Answer 1

Does this seem sound?

Answer 2

o..m..g..

Answer 3

I agree with Hershey_Kisses...

Answer 4

holy cow! that looks like.....holy cow that unbearable!