Theorem: Let $a_{n}=a_{n-1}+1, a_1=1$. For any $n$, in order to compute $a_n$, it is necessary to compute $a_i$ for each $i=1,\dots,n-1$, which takes $\Theta(n)$ time.
Proof: This is vacuously true for $n=1$. Assume true for $n=k-1$. Prove true for $n=k$. In order to compute $a_{k-1}+1$, it is necessary to compute $a_{k-1}$. Then since $a_k=a_{k-1}+1$, in order to compute $a_k$, it is necessary to compute $a_{k-1}$. By the induction hypothesis, in order to compute $a_{k-1}$, it is necessary to compute $a_i$ for each $i=1,\dots,k-2$. Hence, in order to compute $a_k$, it is necessary to compute $a_i$ for each $i=1\dots,k-1$. QED
What is wrong with this proof? It seems valid to me, even though the theorem is false.
In your induction step, you made the additional assumption (beyond the inductive hypothesis) that it was necessary to compute $a_{k-1}$ in order to compute $a_k$. That's hardly the case, as simple inspection of the recursive definition gives us the closed-form definition $a_n:=n$ for all $n$.