I had a quick question:

Is the following proof of the theorem below correct?

Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1.

Proof: Since C is convex, then

t*x + (1-t)*y is contained in C, for every x,y in C, and for 0<=t<=1.

Since y = 0 is contained in C, then in particular,

t*x = t*x + (1-t)*0 = t*x + (1-t)*y is contained in C, for every x in C, and for 0<=t<=1.

Hence tC is a subset of C for 0<=t<=1. This completes the proof.

Is the above proof correct? Or, did I make a mistake in the proof?

Is the following proof of the theorem below correct?

Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1.

Proof: Since C is convex, then

t*x + (1-t)*y is contained in C, for every x,y in C, and for 0<=t<=1.

Since y = 0 is contained in C, then in particular,

t*x = t*x + (1-t)*0 = t*x + (1-t)*y is contained in C, for every x in C, and for 0<=t<=1.

Hence tC is a subset of C for 0<=t<=1. This completes the proof.

Is the above proof correct? Or, did I make a mistake in the proof?

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