If I can prove f(n) = g(n+1) by induction when n is finite, Can I prove f(n) = g(n) by taking n = $\infty$

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I have to prove f(n) = g(n) when $n = \infty$. Now I can prove f(n) = g(n+1) by induction when n is finite. Can I say $f(n) = g(n)$ by taking $n = \infty$?