Would this argument be valid for proving A for all real numbers greater than or equal to a:
Prove that A is true at n = a.
Assume that A is true for all $ a\le n < k$. Prove that A is true at k.
Would this argument be valid for proving A for all real numbers greater than or equal to a:
Prove that A is true at n = a.
Assume that A is true for all $ a\le n < k$. Prove that A is true at k.
No, because your inductive step assumes that you know $A$ holds on some initial interval of the form $[a,k)$, but you have only shown that $A$ holds at a single initial point.