Let $1 \leq p < q \leq \infty$ ($p$ and $q$ are not related) conclude that the identity map I : $ l^n_p → l^n_q$ has operator norm exactly 1.
I figured I need to show that given $\|Ix\| \leq c\|x\|$ from the wiki definition the infimum over c is 1 so that $\|Ix\| = \|x\|$. is this a correct definition for the operator norm? and is it a good approach?
Can i apply Riesz–Thorin theorem here? if so how?
What you need to show is that $\|Ix\|_q \le \|x\|_p$ for all $x \in \ell_p$ and that the ratio $\dfrac{\|Ix\|_q}{\|x\|_p}$ can be made arbitrarily close to $1$.
The second point is easy. If a sequence $x$ has exactly one nonzero term then $\|Ix\|_q = \|x\|_p$.
On the other hand if you write $\|x\|_\infty$ for the maximum of the terms of a sequence $x \in \ell_r$ then $$\|Ix\|_q^q = \sum_k |x_k|^q \le \|x\|_\infty^{q-p} \sum_k |x_k|^p = \|x\|_\infty^{q-p} \|x\|_p^p \le \|x\|_p^q.$$ Now take the $q$-root on both sides.