Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \ell^p$. Let $\lambda, \mu \in \R$. Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a real-valued function. Let us use real number addition and multiplication. Define pointwise addition as: :$\sequence {a_n + b_n}_{n \mathop \in \N} := \sequence {a_n}_{n \mathop \in \N} +_\R \sequence {b_n}_{n \mathop \in \N}$. Define pointwise scalar multiplication as: :$\sequence {\lambda \cdot a_n}_{n \mathop \in \N} := \lambda \times_\R \sequence {a_n}_{n \mathop \in \N}$ Let the additive inverse be $\sequence {-a_n} := - \sequence {a_n}$. === Closure Axiom === By assumption, $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N} \in \ell^p$. By definition: :$\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}^p < \infty$ :$\displaystyle \sum_{n \mathop = 1}^\infty \size {b_n}^p < \infty$ Consider the sequence $\sequence {a_n + b_n}$. Then:  {{begin-eqn}} {{eqn| l = \sum_{n \mathop = 1}^\infty \size {a_n + b_n}^p      | o = \le      | r = \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }^p }} {{eqn| o = \le      | r = \sum_{n \mathop = 1}^\infty \paren {\map \max {\size {a_n}, \size {b_n} } + \map \max {\size {a_n}, \size {b_n} } }^p      | c = {{defof|Max Operation}} }} {{eqn| r = \sum_{n \mathop = 1}^\infty 2^p \paren {\map \max {\size {a_n}, \size {b_n} } }^p }} {{eqn | o = \le       | r = 2^p \sum_{n \mathop = 1}^\infty \paren {\size {a_n}^p + \size {b_n}^p} }} {{eqn | o = <       | r = \infty       | c = $\sequence {a_n}, \sequence {b_n} \in \ell^p$ }} {{end-eqn}} Hence: :$\sequence {a_n + b_n} \in \ell^p$ {{qed|lemma}} === Commutativity Axiom === By Pointwise Addition on Ring of Sequences is Commutative, $\sequence {a_n} + \sequence {b_n} = \sequence {b_n} + \sequence {a_n}$ {{qed|lemma}} === Associativity Axiom === By Pointwise Addition on Ring of Sequences is Associative, $\paren {\sequence {a_n} + \sequence {b_n} } + \sequence {c_n} = \sequence {a_n} + \paren {\sequence {b_n} + \sequence {c_n} }$. {{qed|lemma}} === Identity Axiom === {{begin-eqn}} {{eqn | l = \sequence {0 + a_n}       | r = \sequence 0 +_\R \sequence {a_n}       | c = {{Defof|Pointwise Addition on Ring of Sequences}} }} {{eqn | r = \tuple {0, 0, 0, \dots} +_\R \sequence {a_n}       | c = Definition of $\sequence 0$ }} {{eqn | r = \sequence {a_n} }} {{end-eqn}} {{qed|lemma}} === Inverse Axiom === {{begin-eqn}} {{eqn | l = \sequence {a_n + \paren {-a_n} }       | r = \sequence {a_n} +_\R \sequence {-a_n}       | c = {{Defof|Pointwise Addition on Ring of Sequences}} }} {{eqn | r = \sequence {a_n} +_\R \paren {-1} \times_\R \sequence {a_n}        | c = Definition of $\sequence {-a_n}$ }} {{eqn | r = 0 }} {{end-eqn}} {{qed|lemma}} === Distributivity over Scalar Addition === {{begin-eqn}} {{eqn | l = \sequence {\paren {\lambda +_\R \mu} \cdot a_n }       | r = \paren {\lambda +_\R \mu} \times_\R \sequence {a_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \lambda \times_\R \sequence {a_n} +_\R \mu \times_\R \sequence {a_n}       | c = Real Multiplication Distributes over Addition }} {{eqn | r = \sequence {\lambda \cdot a_n} +_\R \sequence {\mu \cdot a_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \sequence {\lambda \cdot a_n + \mu \cdot a_n}       | c = {{Defof|Pointwise Addition on Ring of Sequences}} }} {{end-eqn}} {{qed|lemma}} === Distributivity over Vector Addition === {{begin-eqn}} {{eqn | l = \lambda \times_\R \sequence {a_n + b_n}       | r = \lambda \times_\R \paren {\sequence {a_n} +_\R \sequence {b_n} }       | c = {{Defof|Pointwise Addition on Ring of Sequences}} }} {{eqn | r = \lambda \times_\R \sequence {a_n} +_\R \lambda \times_\R \sequence {b_n}       | c = Real Multiplication Distributes over Addition }} {{eqn | r = \sequence {\lambda \cdot a_n} +_\R \sequence {\lambda \cdot b_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \sequence {\lambda \cdot a_n + \mu \cdot b_n}       | c = {{Defof|Pointwise Addition on Ring of Sequences}} }} {{end-eqn}} {{qed|lemma}} === Associativity with Scalar Multiplication === {{begin-eqn}} {{eqn | l = \sequence {\paren {\lambda \times_\R \mu} \cdot a_n}       | r = \paren {\lambda \times_\R \mu} \times_\R \sequence {a_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \lambda \times_\R \paren {\mu \times_\R \sequence {a_n} }       | c = Real Multiplication is Associative }} {{eqn | r = \lambda \times_\R \sequence {\mu \cdot a_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \sequence {\lambda \cdot \paren {\mu \cdot a_n} }       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{end-eqn}} {{qed|lemma}} === Identity for Scalar Multiplication === {{begin-eqn}} {{eqn | l = \sequence {1 \cdot a_n}       | r = 1 \times_\R \sequence {a_n}       | c = {{Defof|Pointwise Scalar Multiplication on Ring of Sequences}} }} {{eqn | r = \sequence {a_n} }} {{end-eqn}} {{qed|lemma}} {{qed}}