Let $\displaystyle {x}=[x_{1}, x_{2}, ... ,x_{n}]$ be a vector in $\displaystyle \mathbb{R}^{n}$. Prove that $\displaystyle ||x|| \leq\sum_{i=1}^n |x_{i}|$.
I am not sure how to start this. I'm guessing I need to use induction but I don't know how.
Let $\displaystyle {x}=[x_{1}, x_{2}, ... ,x_{n}]$ be a vector in $\displaystyle \mathbb{R}^{n}$. Prove that $\displaystyle ||x|| \leq\sum_{i=1}^n |x_{i}|$.
I am not sure how to start this. I'm guessing I need to use induction but I don't know how.
Let $\displaystyle a=x^2,b=y^2$ where $\displaystyle x,y>0$.
Therefore, you want to prove $\displaystyle \sqrt{x^2+y^2} \leq \sqrt{x^2} + \sqrt{y^2} = x + y$.
We will prove this in a geometric way. Imagine a right triangle with legs $\displaystyle x,y$. The third leg is $\displaystyle \sqrt{x^2+y^2}$ by Pythagorean theorem. However, by the triangle inequality it means the sum of any two sides must be great than the third sides, therefore, $\displaystyle \sqrt{x^2+y^2}\leq x+y$. Can you prove your vector inequality now?
Yes, that makes sense to me, and I was able to prove the rest. Thank you.
I am having difficulty with the proofs in this class :/ I have never had to do proofs before. I can do them for the homework by studying the book and getting help from others, but today I had a test and didn't do well. There were 4 proofs, and I was only able to do two and most of the third, before running out of time.
Do you know of a book or some other resource to help me get more practice? My book has some proofs (not many), yet my professor is almost entirely focused on proofs. I can understand when she writes them on the board but that is completely different from inventing my own.