I encountered such an exercise:
Let $n,m$ be two positive integers with $n\geq 2m$ and $M$ be a smooth manifold of dimension $m$. Let $p,q$ be two distinct points on $M$. Show that there exists a smooth immersion $h:M\to \mathbb R^n$ such that $h(p)= h(q)$.
The Whitney Immersion Theorem tells us:
If $M$ is a smooth manifold of dimension $m$, and $n\geq 2m$. Then for any $f\in C^\infty(M,\mathbb R^n)$ and any $\varepsilon>0$, we can find a $g\in C^\infty (M,\mathbb R^n)$ such that (1) $g$ is a smooth immersion from $M$ into $\mathbb R^n$; (2) $\|g(x)-f(x)\|_{\mathbb R^n}<\varepsilon$ for any $x\in M$.
Clearly, the Witney Immersion Theorem guarantees the existence of a smooth immersion $h:M\to \mathbb R^n$ (since $f:M\to\mathbb R^n$ defined by $f\equiv 0$ is smooth), but how can we construct a smooth immersion $h$ that satisfies the additional condition $h(p)=h(q)$? Any ideas would be appreciated.