fa.functional analysis – An orthogonal projection sequence in the Hilbert space

Let $ H $ to be an infinite dimensional space of Hilbert on $ mathbb {C} $

Let $ {v_n } _ {n in mathbb {N}} subset H $ to be a linearly independent sequence of vectors in $ H $ such as $ v_n to u $

Let $ forall m in mathbb {N}: V_m = operatorname {span} {v_n } _ {n geq m} $ and $ P_m $ to be the orthogonal projection on $ V_m $

My question is to know if it is true that:
$$
forall v in V_1:
lim_ {m to infty}
P_m (v) = a cdot u
$$

in $ H $-norm and with $ a in mathbb {C} $

Thank you.