Set Point Topology-product and connected spaces

1. Prove that the space C([0,1]) is arc-connected (C([0,1]) = real continous functions onto [0,1] with the metric max|f(x)-g(x)| )

2. Prove that in a product space of infinite many spaces, such as in each space there is more than one point, every point is an accumulation point.

3. Prove that $\displaystyle \mathbb{R}_{CF}$ is path-connected but not arc-connected.

I'm realy bad at all the path-connected&arc-connected subject so I can't realy understand how I should start solving these 3 problems...I have no clue about them...

Tnx in advance