Csdrhrt

Let $mathcal{H}$ denote a separable Hilbert space, with orthonormal basis ${v_k}_{k=1}^infty$. Let $N in mathbb{N}$, and consider the subspace $M := text{span} {v_k}_{k=1}^infty$.

Show that for any $f in mathcal{H}$ and any coefficients $c_1,...,c_N in mathbb{C}$,
begin{equation*}
left| f- sum_{k=1}^N c_kv_k right|^2 = sum_{k=1}^N |c_k - langle f,v_k rangle |^2 + sum_{k=N+1}^infty |langle f,v_k rangle |^2
end{equation*}

My attempt so far
begin{align}
left| f- sum_{k=1}^N c_kv_k right|^2 & = leftlangle f- sum_{k=1}^N c_kv_k, f- sum_{k=1}^N c_kv_k rightrangle \
& = leftlangle f, f- sum_{k=1}^N c_kv_k rightrangle -sum_{k=1}^N c_k leftlangle v_k, f- sum_{k=1}^N c_kv_k rightrangle \
& = langle f,f rangle - sum_{k=1}^N overline{c}_k langle f,v_k rangle - sum_{k=1}^N overline{c}_k langle v_k,f rangle + sum_{k=1}^N sum_{l=1}^N overline{c}_l c_k langle v_k,v_l rangle
end{align}
I'm not really sure if I'm on the right track. Can somebody help me with the next step. Thanks.

Since ${v_k}_{k=1}^infty$ is an orthonormal basis of $mathcal H$, Parseval's identity shows that for any vector $u$, the following equality holds:
$$
leftlVert urightrVert^2=sum_{k=1}^{+infty}leftlvert leftlangle u,v_krightranglerightrvert^2.
$$
Apply this to $u:= f-sum_{i=1}^N c_iv_i$. Then for all $kgeqslant 1$,
$$
leftlangle u,v_krightrangle=leftlangle f,v_krightrangle
-sum_{i=1}^N c_ileftlangle v_i,v_krightrangle.
$$
If $1leqslant kleqslant N$, then $sum_{i=1}^N c_ileftlangle v_i,v_krightrangle=c_k$ hence $leftlvert leftlangle u,v_krightranglerightrvert^2=leftlvert leftlangle f,v_krightrangle-c_krightrvert^2$ and if $kgeqslant N+1$, then $sum_{i=1}^N c_ileftlangle v_i,v_krightrangle=0$ hence
$leftlvert leftlangle u,v_krightranglerightrvert^2=leftlvert leftlangle f,v_krightrangle rightrvert^2$.