This is not a homework.
Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like:
Let $\chi$ be an irreducible $\mathbb{F}$-character of a finite group $G$ with degree $n$, where $\mathbb{F}$ is an algebraically closed field of characteristic zero. If $H$ is a subgroup of $G$, show that there is an irreducible $\mathbb{F}$-character with degree at least $\frac{n}{|G:H|}$.
I think that I should use induced character and Frobenius Reciprocity Theorem to build this character but unfortunately, all my constructed characters didn't satisfy (or I don't no it satisfied or not) the conditions of this statement.
I will be so appreciate for any helpful answer.
Hint: if $\chi \in Irr (G)$, and $\phi$ is an irreducible constituent of $\chi_H$ (so $[\chi_H,\phi] \geq 1$), then by Frobenius Reciprocity $\phi^G(1)=\phi(1)[G:H] \geq \cdots + \chi(1) + \cdots \geq n$.