Discrete Cauchy-Schwarz Proof

Question

I am wondering if someone could help me with part (c) as I'm not sure how to approach it. I have also attached my solutions to parts (a) and (b) and would be thankful if someone could verify if what I have done is correct (especially (a) where I just assumed the ds changes to dx at the end).

Cauchy Problem (c)

Solutions for (a) and (b)

Answer 1

If $v(0) = 0$, with $u(x) = v(x)^2$ then applying Cauchy-Schwarz $\sup_{x \in [0,1]} v(x)^2 \le \int_0^1 |u'(x)|dx = 2\int_0^1 |v(x)v'(x)|dx \le 2 (\int_0^1 |v(x)|^2dx)^{1/2}(\int_0^1 |v'(x)|^2dx)^{1/2}$.

The discrete analogue is :

If $v(0) = 0$ : $\sup_{n \ge 1} v(n)^2 \le \sum_{n \ge 1} |v(n)^2-v(n-1)^2| = \sum_{n \ge 1} |v(n)-v(n-1)|\, |v(n)+v(n-1)|$ $ \le (\sum_{n \ge 1} |v(n)+v(n-1)|^2)^{1/2}(\sum_{n \ge 1} |v(n)-v(n-1)|^2)^{1/2}$