1) What does it mean that f is continuous? What happens to sequences that converge in when they are passed through f? So, you want to find a sequence x_{(n,1)} that converges very close to zero such that converges very close to L. Then you want a sequence of sequences that converges closer and closer to 0 as m and n approach infinity such that approaches closer and closer to L as m and n approach infinity. Use the results of liminf and limsup.

2) I would need to think about it.