Let $A,B$ $\in \mathbb{R} ^{n\times n}$. For any induced norm $\left\lVert \cdot\right\rVert$ on $\mathbb {R}^{n\times n}$ show $\left\lVert (AB)\right\rVert \leq \left\lVert A\right\rVert \left\lVert B\right\rVert.$
$$\left\lVert (AB)_{ij}\right\rVert =\left\lVert \sum_{k=1}^n [A]_{ik}.[B]_{kj}\right\rVert=\sqrt{\sum_{k=1}^n\bigr([A]_{ik}.[B]_{kj}\bigr)^2}\leq \sqrt{\sum_{k=1}^n\bigr([A]_{ik}\bigr)^2}\sqrt{\sum_{k=1}^n\bigr([B]_{ik}\bigr)^2}=\left\lVert A_{ik}\right\rVert \left\lVert B_{kj}\right\rVert$$
here $1\leq i \leq n, 1\leq j \leq n$
There are other proofs which I can't understand. I was wondering if this proof will suffice or am I missing something ?
Your proof doesn't seem to make sense. You confuse scalars, vectors and matrices.
As $\|.\|$ is an induced norm, there exists a norm $\|.\|'$ on $\Bbb R^n $ such that $$\|M\|=\sup_{x\neq 0}\frac{\|Mx\|' }{\|x\|'}.$$ Now, for every $x\neq 0$, we have $$\|ABx\|' = \|Ay\|'\leq \|A\|\|Bx\|'\leq \|A\|\|B\|\|x\|'.$$ It follows $\|AB\|\leq \|A\|\|B\|$.