This is the question:
Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur.
I thought about the immersion $i:\mathbb{S}^2 \to \mathbb{R}^3$, but I only have some geometrical insight, I don't know how to prove analytically. Please help me - thanks.