#Java, 85 bytes

<!-- language-all: lang-java -->

This is surprisingly short in Java.

    void a(int a){String b="";for(int c=0;c++<a;System.out.println(b))b+=(c>1?" ":"")+c;}

Indented:

    void a(int a){
        String b="";
        for(int c=0;c++<a;System.out.println(b))
            b+=(c>1?" ":"")+c;
    }