A Banach space is a normed vector space, where a Cauchy sequence converges {{WRT}} the supplied norm. To prove the theorem, we need to show that a Cauchy sequence in $\struct {\ell^p, \norm {\,\cdot\,}_p}$ converges. We take a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {\ell^p, \norm {\,\cdot\,}_p}$. Then we consider the $k$th component and show, that a real Cauchy sequence $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$ with the limit $x^{\paren k}$ and denote the entire set as $\mathbf x$. Finally, we show that $\sequence {\mathbf x_n}_{n \in \N}$, composed of components $x_n^{\paren k},$ converges in $\struct {\ell^p, \norm {\,\cdot\,}_p}$ with the limit $\mathbf x$. Let $\sequence {\mathbf x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {\ell^p, \norm{\, \cdot \,}_p}$. Denote the $k$th component of $\mathbf x_n$ by $x_n^{\paren k}$. === $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$===  Let $\epsilon >0$. Then: :$\displaystyle \exists N \in \N : \forall m,n \in \N : m,n > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$ For same $N, m, n$ consider $\size {x_n^{\paren k} - x_m^{\paren k} } $: {{begin-eqn}} {{eqn | l = \size {x_n^{\paren k} - x_m^{\paren k} }        | r = \paren {\size {x_n^{\paren k} - x_m^{\paren k} }^p}^{\frac 1 p} }} {{eqn | o = \le       | r = \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p}^{\frac 1 p} }} {{eqn | r = \norm {\mathbf x_n - \mathbf x_m}_p       | c = {{defof|P-Norm}} }} {{eqn | o = <       | r = \epsilon }} {{end-eqn}} Hence, $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ is a Cauchy sequence in $\struct {\R, \size {\, \cdot \,}}$. From Real Number Line is Complete Metric Space, $\R$ is a complete metric space. Consequently, $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$. {{qed|lemma}} Denote the limit $\displaystyle \lim_{n \mathop \to \infty} \sequence {x_n^{\paren k}}_{n \mathop \in \N} = x^{\paren k}$. Denote $\sequence {x^{\paren k}}_{k \mathop \in \N} = \mathbf x$. === $\mathbf x$ belongs to $\ell^p$ === Let $\epsilon > 0$. Then: :$\exists N \in \N : \forall n,m \in \N : n,m > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$. Let $K \in \N$, $1 \le p < \infty$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^K \size {x_n^{\paren k} - x_m^{\paren k} }^p        | o = \le       | r = \sum_{k \mathop = 1}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p   }} {{eqn | r = \norm {\mathbf x_n - \mathbf x_m}_p^p }} {{eqn | o = <       | r = \epsilon^p }} {{eqn | o = <       | r = \infty  }} {{end-eqn}} Take the limit $m \to \infty$:  :$\displaystyle \sum_{k \mathop = 1}^K \size {x_n^{\paren k} - x^{\paren k}}^p < \infty$ Since $K$ was arbitrary, we can take the limit $K \to \infty$. By definition, $\forall k \in \N : x_n^{\paren k} - x^{\paren k} \in \ell^p$. In other words, $\mathbf x_n - \mathbf x \in \ell^p$. But, by assumption, $\mathbf x_n \in \ell^p$. Therefore: :$\paren {\mathbf x - \mathbf x_n} + \mathbf x_n = \mathbf x \in \ell^p$ {{qed|lemma}} === $\sequence {\mathbf x_n}_{n \mathop \in \N}$ converges in $\struct {\ell^p, \norm {\, \cdot \,}_p}$ to $\mathbf x$=== Let $1 \le p < \infty$.  Let $\epsilon > 0$. Fix $N \in \N$, $m > N$. Then: :$\displaystyle \forall n > N : \norm {\mathbf x_n - \mathbf x_m}_p < \epsilon$ We have that $\sequence {x_n^{\paren k}}_{n \mathop \in \N}$ converges in $\struct {\R, \size {\, \cdot \,}}$. Take the limit $m \to \infty$: {{begin-eqn}} {{eqn | l = \lim_{m \mathop \to \infty} \norm {\mathbf x_n - \mathbf x_m}_p       | r = \lim_{m \mathop \to \infty} \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x_m^{\paren k} }^p }^{\frac 1 p} }} {{eqn | r = \paren {\sum_{k \mathop = 0}^\infty \size {x_n^{\paren k} - x^{\paren k} }^p }^{\frac 1 p}       | c = Limit of Composite Function }} {{eqn | r = \norm {\mathbf x_n - \mathbf x}_p }} {{eqn | o = <       | r = \epsilon }} {{end-eqn}} So $\sequence {\mathbf x_n}_{n \mathop \in \N}$ converges in $\struct {\ell^p, \norm {\, \cdot \,}_p}$. {{qed}}