Help with Norms

Let $x \in \mathbb{R}^n$, and for notation simplicity, let's assume that $x_i \geq 0$ for $1 \leq i \leq n$ (it is enough because all the written norms only depend on $\lvert x_i \rvert$ rather than $x_i$ itself).

The desired constants are $c_1 = \sqrt{n}$, $c_2 = n$, and $c=1$. Let us show that.

For $c_1$, we have \[ \lVert x \rVert_1 = \sum_i \lvert x_i \rvert = \sqrt{\left( \sum_i \lvert x_i \rvert \right)^2} = \sqrt{\sum_i \lvert x_i \rvert^2 + \sum_{i<j} 2 \lvert x_i \rvert \lvert x_j \rvert } \leq \sqrt{\sum_i \lvert x_i \rvert^2 + \sum_{i<j} \left( \lvert x_i \rvert^2 + \lvert x_j \rvert^2 \right) } \] where I used the fact that $a^2 + b^2 \geq 2ab$. Now, \[ \sqrt{\sum_i \lvert x_i \rvert^2 + \sum_{i<j} \left( \lvert x_i \rvert^2 + \lvert x_j \rvert^2 \right) } = \sqrt{n \sum_i \lvert x_i \rvert^2} = \sqrt{n} \sqrt{\sum_i \lvert x_i \rvert^2} = \sqrt{n} \lVert x \rVert_2 \] as expected.

For $c_2$, we have \[ \lVert x \rVert_1 = \sum_i \lvert x_i \rvert \leq n \max \{ \lvert x_i \rvert \mid 1 \leq i \leq n \} = n \lVert x \rVert_{\infty} \] as expected.

For $c$, we have \[ \lVert x \rVert_2 = \sqrt{\sum_i \lvert x_i \rvert^2} \leq \sum_i  \sqrt{\lvert x_i \rvert^2} = \sum_i \lvert x_i \rvert = \lVert x \rVert_1 \] where I used the fact that, for positive $a, b$, we have $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$ (as taking the square gives the trivial inequality $a+b \leq a+b+2\sqrt{ab}$). The equality holds when all but one coordinate of $x$ are $0$, so for example $x = (1, 0, \dots, 0)$, for which both the norms are equal to $1$.