How to prove the boundedness of a sequence?

Let $\{q_n\}_n$ be a divergent sequence in $\mathbb{R}^2$ and $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function such that $\lim_{n\rightarrow\infty}f(q_n)=\inf_{\mathbb{R}^2}f$. For a fixed point $o\in \mathbb{R}^2$, suppose $\gamma_n:[0,dist(o,q_n)]\rightarrow \mathbb{R}^2$ is a arc length parametrized curve from $o$ to $q_n$ for each $n$. Now I want to prove that the sequence $\{(f\circ\gamma_n)'(0)\}_n$ is bounded.
Please someone help me in proving this.
Thank you.

$$||(f\circ\gamma_n)^\prime(0)|| =|\langle\nabla f(o),\gamma_n^\prime(0)\rangle|\le ||\nabla f(o)||$$ since, by assumption, $\gamma_n^\prime(0)$ has length $1$. Am I missing something?