document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166341>Question 1166341</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let W be the set of all sequences which converge to 0. Is W a subspace of the vector space of convergent sequences? You must justify your answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852765 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852765\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Yes, **W**, the set of all sequences which converge to $0$, is a **subspace** of the vector space of convergent sequences, which we can call $V$.\r<BR>\n" );
document.write( "\n" );
document.write( "The vector space $V$ is the set of all sequences of real numbers, $\{a_n\}_{n=1}^\infty$, such that $\lim_{n \to \infty} a_n = L$, where $L$ is a finite real number.<BR>\n" );
document.write( "The subset $W$ is defined as:<BR>\n" );
document.write( "$$W = \left\{ \{a_n\} \in V \ \bigg| \ \lim_{n \to \infty} a_n = 0 \right\}$$\r<BR>\n" );
document.write( "\n" );
document.write( "To prove that $W$ is a subspace of $V$, we must verify the three subspace conditions:\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 1. The Zero Vector Condition (Contains the Zero Sequence)\r<BR>\n" );
document.write( "\n" );
document.write( "The zero vector in $V$ is the **zero sequence**, $\mathbf{z} = \{z_n\}$, where $z_n = 0$ for all $n \ge 1$.\r<BR>\n" );
document.write( "\n" );
document.write( "We check if the zero sequence satisfies the condition for $W$:<BR>\n" );
document.write( "$$\lim_{n \to \infty} z_n = \lim_{n \to \infty} 0 = 0$$<BR>\n" );
document.write( "Since the limit of the zero sequence is $0$, the zero sequence is in $W$.<BR>\n" );
document.write( "Thus, $W$ is non-empty.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 2. Closure under Vector Addition ➕\r<BR>\n" );
document.write( "\n" );
document.write( "Let $\mathbf{a} = \{a_n\}$ and $\mathbf{b} = \{b_n\}$ be two arbitrary sequences in $W$. This means:<BR>\n" );
document.write( "* $\lim_{n \to \infty} a_n = 0$<BR>\n" );
document.write( "* $\lim_{n \to \infty} b_n = 0$\r<BR>\n" );
document.write( "\n" );
document.write( "We need to check if their sum, $\mathbf{a} + \mathbf{b} = \{a_n + b_n\}$, is also in $W$. We use the **Limit Sum Law**:<BR>\n" );
document.write( "$$\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n$$<BR>\n" );
document.write( "Substitute the conditions for $\mathbf{a}$ and $\mathbf{b}$:<BR>\n" );
document.write( "$$\lim_{n \to \infty} (a_n + b_n) = 0 + 0 = 0$$<BR>\n" );
document.write( "Since the sum of the sequences converges to $0$, the sequence $\mathbf{a} + \mathbf{b}$ is in $W$.<BR>\n" );
document.write( "Thus, $W$ is closed under sequence addition.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 3. Closure under Scalar Multiplication ✖️\r<BR>\n" );
document.write( "\n" );
document.write( "Let $\mathbf{a} = \{a_n\}$ be an arbitrary sequence in $W$ (so $\lim_{n \to \infty} a_n = 0$), and let $c$ be an arbitrary scalar (a real number).\r<BR>\n" );
document.write( "\n" );
document.write( "We need to check if the scalar multiple, $c\mathbf{a} = \{c \cdot a_n\}$, is also in $W$. We use the **Limit Constant Multiple Law**:<BR>\n" );
document.write( "$$\lim_{n \to \infty} (c \cdot a_n) = c \cdot \lim_{n \to \infty} a_n$$<BR>\n" );
document.write( "Substitute the condition for $\mathbf{a}$:<BR>\n" );
document.write( "$$\lim_{n \to \infty} (c \cdot a_n) = c \cdot (0) = 0$$<BR>\n" );
document.write( "Since the scalar multiple of the sequence converges to $0$, the sequence $c\mathbf{a}$ is in $W$.<BR>\n" );
document.write( "Thus, $W$ is closed under scalar multiplication.\r<BR>\n" );
document.write( "\n" );
document.write( "***\r<BR>\n" );
document.write( "\n" );
document.write( "Since $W$ satisfies all three conditions, it is a **subspace** of the vector space of convergent sequences, $V$.\r<BR>\n" );
document.write( "\n" );
document.write( "Would you like to analyze a different set of sequences, such as the set of sequences that converge to $1$, to see if it is a subspace? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );