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Let $X$ be a Normed Linear Space and $M$ be a closed subspace of X. Assume that both $M$ and $X/M$ are banach spaces. Prove that $X$ is a banach space.

So firstly I assumed a Cauchy sequence ${(f_n)}$ in $X$, then ${(f_n + M)}$ is also cauchy in $X/M$. Suppose it converges to $f+M$. Then $||(f_n-f)+M||$ tends to 0 as n tends to infinity. So by the property of this norm there exists a sequence $(g_n)$ in $M$ such that $(f_n+g_n)$ converges to $f$. So I have to show that $(g_n)$ is cauchy and hence $(f_n)$ converges. But I'm facing problem in showing that $(g_n)$ is cauchy.

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You know that $f_n + g_n$ is Cauchy in $X$ since it converges to $f$ in $X$ by the choice of $g_n$. By assumption, you also know that $f_n$ is a Cauchy sequence. It then follows that $g_n = (f_n + g_n) - f_n$ is a Cauchy sequence as it is a sum of two Cauchy sequences.

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  • $\begingroup$ Haha I'm a fool. Thank you btw. $\endgroup$ – MathCosmo May 10 at 14:47

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