Fixed point question with convergence

Question

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ is $C^1$ and $1$ to $1$ and there exists a strict increasing sequence $t_{n} \in \mathbb{N}$ s.t $f^{t_{n}}(x) \to p$ for all $x$ as $n\to \infty$ (composition $t_{n}$ times) where $p$ is a fixed point, i.e $f(p)=p$. Can you prove or find a counterexample that $f^n(x) \to p$ as $n \to \infty$ for all $x$.