std::vector initializations and typedefs

I know how to do this the long way so to speak:

#include <vector>

int main() {
    // Simple vector of ints = resized to 1k elements
    std::vector<int>  ints;
    ints.resize( 1000 );  // Easy enough

    // nested vector of ints 1k vectors each with 1k elements
    std::vector<std::vector<ints>> vecInts;
    vecInts.resize( 1000 );
    for ( auto& a : vecInts ) {
        a.resize( 1000 );
    }
    // Again easy enough.
}

Now instead of typing it out like that I would like to use typedefs

#include <vector>

typedef std::vector<int> Ints;
typedef std::vector<Ints> vecInts;   

int main() {
    vecInts a;

    a.resize( 1000 ); // seems okay
    // Assuming this would work...
    for ( auto& n : a ) {
        n.resize( 1000 );
    }
}

My question is does the 2^nd code snippet do what is expected and is it equivalent to the 1^st code snippet or am I missing something?

Second quick question does 1k * 1k exceed the size limits of std::vector?