Can all subsequences of a function sequence in C([0,1], R) converge uniformly?

[Malmstrom]

Let y_n be a sequence of functions in \mathcal{C}([0,1], \mathbb{R})
Suppose that every subsequence of y_n has a subsequence that converges uniformly. Prove that they all converge to the same limit.

 

[Landau]

So, what have you tried?

 

[Malmstrom]

This is not an exercise. This is a question that rose solving an exercise. Can't solve it.