- #1
moweee
- 3
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Hello all.
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 {an} converges to a in the standard topology, denoted by τs, then {can} converges to ca in the standard topology on ℝ.
Proof
Let c [itex]\in[/itex] ℝ such that c≠0. Suppose {an} converges to a in the standard topology, denoted by τs.
Let V [itex]\in[/itex] τs with ca in V. Since V in τs, 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] τs such that a [itex]\in[/itex] (p/c, q/c).
Since, by our assumption, {an} converges to a in the standard topology, there exists m [itex]\in[/itex] N such that an [itex]\in[/itex] (p/c,q/c) for all n ≥ m. Hence, p/c < an < q/c. Hence, p < can< q for all n≥m. Since can [itex]\in[/itex] (p,q) and (p,q) [itex]\subseteq[/itex] V, can [itex]\in [/itex]V for all n ≥ m.
Therefore, {can} converges to ca in the standard topology on ℝ.
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 {an} converges to a in the standard topology, denoted by τs, then {can} converges to ca in the standard topology on ℝ.
Proof
Let c [itex]\in[/itex] ℝ such that c≠0. Suppose {an} converges to a in the standard topology, denoted by τs.
Let V [itex]\in[/itex] τs with ca in V. Since V in τs, 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] τs such that a [itex]\in[/itex] (p/c, q/c).
Since, by our assumption, {an} converges to a in the standard topology, there exists m [itex]\in[/itex] N such that an [itex]\in[/itex] (p/c,q/c) for all n ≥ m. Hence, p/c < an < q/c. Hence, p < can< q for all n≥m. Since can [itex]\in[/itex] (p,q) and (p,q) [itex]\subseteq[/itex] V, can [itex]\in [/itex]V for all n ≥ m.
Therefore, {can} converges to ca in the standard topology on ℝ.