my teacher of Mathematical Analysis gave me these two problems:

1) let f be a continuous function on (0,1) and such that liminf f(x) < limsup f(x) for x that approaches 0, then for every real value L that belongs to (liminf f(x), limsup f(x)) exists {Xn} convergent sequence to 0 in (0,1) such that lim f(Xn) = L for n that approaches to infinite.

2) let f(n), g(n) be two sequences such that f(n) >= g(n) for every n natural number, is true or false that liminf f(n) >= limsup g(n) ? Give a proof of that.

i don't know how to give a proof of the first fact, about the second one i proved that is true only if the two limits exist but i didn't find a counter-exemple to justify the assumption (so it could be wrong the entire proof).

Do you get any suggestions?