Prove that the set of continuous real-valued functions on the interval $[0,1]$ is a subspace of $\mathbb{R}^{[0,1]}$ and so is closed under addition

Question

I'm pretty confused about how to prove that it is closed under addition.

The way I'm thinking about this is probably very wrong, but I will describe it so that you can appoint where I did got it wrong.

The sum of two continuous real-valued functions on the interval of [0,1] won't necessarily result in a continuous real-valued function on the interval of [0,1], because the sum of two function whose results are 0.8(which is inside the interval established) and 0.9(which is inside the interval established) will return 1.7 right? which is clearly out of the established interval, and that shows us that it isn't closed under addition, and so should no be considered a subspace right?

So, what did I miss?