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 Homework Statement

Show that ##a^\dagger\lvert n \rangle = \sqrt{n+1}\lvert n+1 \rangle##
(From Shankar, Principles of Quantum Mechanics, Chapter 8  The Harmonic Oscillator)
 Homework Equations

##\hat{H}+\frac{1}{2}=aa^\dagger=a^\dagger a##
##\hat{H}\lvert n \rangle = (n+\frac{1}{2})\lvert n \rangle##
I have written the equation, with an unknown constant
$$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$
I then take the adjoint to get
$$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$
I then multiply them to get
$$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1 \rvert n+1 \rangle C_{n+1}^2$$
On the left hand side, since ##aa^\dagger = \hat{H}\frac{1}{2}##, the expression just simplifies to ##\langle n \rvert n \lvert n \rangle##. On the right hand side, since ##\lvert n+1 \rangle## is a normalized state, it just simplifies to ##C_{n+1}^2##. Thus, we arrive at
$$\langle n \rvert n \lvert n \rangle = C_{n+1}^2$$
$$n = C_{n+1}^2$$
$$C_{n+1}=\sqrt{n}$$.
Thus,
$$a^\dagger \lvert n \rangle = \sqrt{n} \lvert n+1 \rangle$$
Which is wrong.
I can't see where i went wrong. Can someone help?
$$a^\dagger \lvert n\rangle = C_{n+1}\lvert n+1 \rangle$$
I then take the adjoint to get
$$\langle n \rvert a = \langle n+1 \rvert C_{n+1}^\text{*}$$
I then multiply them to get
$$\langle n \rvert aa^\dagger \lvert n \rangle = \langle n+1 \rvert n+1 \rangle C_{n+1}^2$$
On the left hand side, since ##aa^\dagger = \hat{H}\frac{1}{2}##, the expression just simplifies to ##\langle n \rvert n \lvert n \rangle##. On the right hand side, since ##\lvert n+1 \rangle## is a normalized state, it just simplifies to ##C_{n+1}^2##. Thus, we arrive at
$$\langle n \rvert n \lvert n \rangle = C_{n+1}^2$$
$$n = C_{n+1}^2$$
$$C_{n+1}=\sqrt{n}$$.
Thus,
$$a^\dagger \lvert n \rangle = \sqrt{n} \lvert n+1 \rangle$$
Which is wrong.
I can't see where i went wrong. Can someone help?