I have an exercise in my textbook where I have to show that if $1 \leq p \leq q \leq \infty$ then $\mathcal l^p \subset \mathcal l^q$ and that for every $\mathbf x \in \mathcal l^p$ $\Vert\mathbf x \Vert_{\mathcal l^q} \leq \Vert \mathbf x \Vert_{\mathcal l^p}$.
I considered first $\mathbf x \in \mathcal l^p$ with $\Vert x \Vert_{\mathcal l^p} =1$ because I think it would be useful but i'm not sure how to use this to solve this.
I have also considered making use of the Minkowski's inequality in $\mathcal l^p$ where for $1 \leq p \leq \infty$ $\mathbf x \in \mathcal l^p$ $$\Vert\mathbf x+\mathbf y \Vert_{\mathcal l^q} \leq \Vert \mathbf x \Vert_{\mathcal l^p}+\Vert\mathbf y \Vert_{\mathcal l^p}$$ but then i'm again stuck because in my case we have $\mathcal l^q$. Any help would be appreciated.
Suppose $\|y\|_p \leq 1$. Then $|y_i| \leq 1$ for all $i$ so $\sum |y_i|^{q} \leq \sum |y_i|^{p}=1$. Now apply this to $y =\frac x {\|x\|_p}$ (assuming $x \neq 0$). This gives the desired inequality.
[You get $\|\frac x {\|x\|_p}\|_q \leq 1$. Just pull $\|x\|_p$ out of the norm in the left side and bring it to the right side].