Let $V$ be a finite dimensional inner product space. Then $V$ has an orthonormal basis.
My approach: We will use induction on the dimension on $V$. This is trivial if $\dim(V)=1$, as in this case take any $x≠0$ on $V$, then $\{x/\Vert x \Vert\}$ is an orthonormal basis of $V$. So let $\dim(V)>1$ and suppose that we have the desired result for all smaller dimensions. Take any proper subspace $U$ of $V$. Then $0<\dim(U)<\dim(V)$. So, $U^⊥$ is also a proper subspace of $V$. Then by induction conditions both $U$ and $U^⊥$ have orthonormal basis say $A$ and $B$ respectively. Now I have to prove $A \cup B$ is an orthonormal basis of $V$.
How I will show this? Please help me .