Of course, "it holds when n=k" is a sufficient condition for "it holds when n=k+ 1

Besides, in your statement,

If n=7 holds, then it is proved that "n=k holds" and "n=k+ 1 also holds",

It can only be said that when n≥7, the proposition holds.

Whether other values of n are true or not cannot be interpreted from the proof process.