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How can I use the principle of mathematical induction to show that $X(n)<4$ for all $n\geq1$ where $X(n)$ is defined as $$X(n)=\begin{cases}\sqrt{1+2X(n-1)}&n\neq1\\1&n=1\end{cases}$$

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Hint: It suffices to note that the range of the function $f: [1,4) \to \Bbb R$ defined by $f(x) = \sqrt{1 + 2x}$ is $[1,3)$.

More specifically: we are showing that $1 \leq X(n) < 4$ for all integers $n\geq 1$, using the fact that $X(n+1) = f(X(n))$ for all $n \geq 1$. The base case is that this holds for $n = 1$. The inductive step is to show that if this holds for $n = k$, then it also holds for $n = k+1$.

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  • $\begingroup$ Okay but how does the proof work using mathematical induction $\endgroup$
    – Wisdom
    Dec 2 '19 at 8:05
  • $\begingroup$ See my latest edit $\endgroup$ Dec 2 '19 at 8:18

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