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I have to present a proof to our Intro to Topology class and I just wanted to make sure I did it right (before I look like a fool up there).

Proposition

Let c be in ℝ such that c≠0. Prove that if {a_{n}} converges to a in the standard topology, denoted by τ_{s}, then {ca_{n}} converges to ca in the standard topology on ℝ.

Proof

Let c [itex]\in[/itex] ℝ such that c≠0. Suppose {a_{n}} converges to a in the standard topology, denoted by τ_{s}.

Let V [itex]\in[/itex] τ_{s}with ca in V. Since V in τ_{s}, there exists an interval (p,q) with ca [itex]\in[/itex] (p,q) and (p,q) [itex]\subseteq[/itex] V. Thus, p < ca < q, which implies p/c < a < q/c.

Note that p/c < (p/c + a)/2 < a < (q/c + a)/2 < q/c

Thus, (p/c , q/c) [itex]\in[/itex] τ_{s}such that a [itex]\in[/itex] (p/c, q/c).

Since, by our assumption, {a_{n}} converges to a in the standard topology, there exists m [itex]\in[/itex] N such that a_{n}[itex]\in[/itex] (p/c,q/c) for all n ≥ m. Hence, p/c < a_{n}< q/c. Hence, p < ca_{n}< q for all n≥m. Since ca_{n}[itex]\in[/itex] (p,q) and (p,q) [itex]\subseteq[/itex] V, ca_{n}[itex]\in [/itex]V for all n ≥ m.

Therefore, {ca_{n}} converges to ca in the standard topology on ℝ.

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# Homework Help: Sequences and convergence in the standard topology

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