Page 10 of Proof By Contradiction
He turns back to the theatre.
‘Right,’ he says. ‘You’ve had one-sided limits this week. You should be comfortable telling me what it means for a limit to exist from the right, from the left, and whether those two have to agree. Today’s about what you do when the function in the middle refuses to behave. You know the side limits. You trap it.’
Trap it.
My pulse takes the word personally. My pulse has the IQ of a golden retriever this week.
He draws three curves—one high, one low, one messy thing in between that dips and wobbles. Labels themg,h,f.
He turns, looks at the theatre.
‘What can I say aboutf?’
Silence. Two hundred freshers are doing the thing where they go very still because they’re hoping eye contact won’t find them. The girl with the highlighters has her hand half-up and half-down, undecided.
I lift mine.
Didn’t plan to. Didn’t mean to. My hand’s in the air above my head like it belongs to somebody braver and I’ve been renting it out.
He sees me. ‘Yes. Back row. Go on.’
The whole theatre turns. Femi makes a small, strangled noise beside me that I’ll pay for later.
‘It’s got nowhere to go, sir,’ I say.
My voice is even. Low. The words of a man answering a technical question in a lecture. The pressure underneath is different, and I’m not hiding it as well as I’d like.
He tilts his head a fraction. ‘Be more specific.’
‘fis trapped between two functions that both agree on where they’re going. Sofhas to end up in the same place. It doesn’t get a say. The bounds squeeze it down till the only value left for the limit offataisL.’ A beat. ‘It’s the squeeze. It’s in the name.’
Somewhere in the front row, a bloke laughs—small, nervous, he thinks I’m being cheeky with the lecturer, and he’s half-right.
Haldrey doesn’t move.
‘And what do we need, for that to work?’
Here. This is the bit.
‘Both bounds have to approach the same value,’ I say. ‘From the right and from the left. If they don’t agree, if one’s coming in from above and the other’s coming in from below and they don’t meet, thenfisn’t trapped, sir, it’s somewhere in the middle of a gap. No limit. You’ve got to give it somewhere to land.’
Approach, from above, from below. Agree, trapped. Somewhere to land.
I’ve selected every single one of those words.
I stop, bite it back, look at him. He’s got the marker in his right hand, held still against the tray at the bottom of the board, left hand flat on the lectern. The tendon at the side of his neck moves once—a swallow, small and disciplined and entirely visible from the back row if you know what you’re looking for. And I’m looking.
He says, ‘That’s correct.’
Two words, Lancashire vowels, unhurried. Thecorrectsits in his throat for a beat longer than it needs to.
‘Technically complete,’ he adds. ‘Informally phrased.’ A pause. ‘Useful instinct.’
He turns back to the board.
My hand comes down. The theatre goes back to its glazed expressions. Femi’s gone completely still beside me. He heard every word underneath the words.
Haldrey writes the theorem on the board in the same tight hand he writes everything in, and then he talks for forty-five minutes about it. My pen stays still. He holds himself like a held breath.
