Repeated eigenvalues of a symmetric matrix

[matqkks]

I have been trying to prove the following result:
If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors.
Is there a simple proof of this result?

 

[Robert1986]

Do you know that symmetric matrices can be (orthogonally) diaganalised?

 

[matqkks]

Do you know that symmetric matrices can be (orthogonally) diaganalised?

That is the result I am trying to prove. Just need to show the result for repeated eigenvalue.